Absolutely, though mostly science fiction. Soviet censors and citizens did not care for fantasy.
Though I'm sure you're probably more concerned about Later Soviet Fiction, rather than Early Soviet Fiction, we'll cover a bit of both
#Early Soviet Fiction (1917-1940ish)
Early Soviet science fiction stuck to "hard fiction"; fiction with high scientific accuracy. (Some today will say that soft fiction is really science fantasy rather than science fiction, but that is beside the point now.) Such authors include Alexander Belyaev, Grigory Adamov, Vladimir Obruchev, and Aleksey Tolstoy.
Let's discuss some of their most famous works and how it looks compared to their Western counterparts.
Professor Dowell's Head by Alexander Belyayev
By 1908, scientists had attempted head transplants in animals. This had inspired Belyayev to write Professor Dowell's Head, a book centered on the idea of providing life support to separated body parts. This was published in 1925 and contains themes of ethics is science. I don't want to spoil the book, in case anyone does have interest in reading it, but the Doctor conducting the life support experiments is not the Protagonist.
This story, though of course coming first, may remind a modern reader of Hollow Man or The Human Centipede; the story of a scientist gone mad and going beyond appropriate ethics.
Amphibian Man by Alexander Belyaev
This book is a science fiction adventure novel about a boy who's life is saved by receiving a transplant of shark gills. The boy must live in an aquatic environment, but is recruited for his superhuman diving abilities. Like Professor Dowell's Head, it explores themes of scientific ethics, and includes some socialist themes. There is a film adaptation, produced in 1962 if you like old films more than old books.
Plutonia by Vladimir Obruchev
Written in 1915, but not published until 1924, Plutonia echoes of Sir Arthur Canon Doyle's The Lost World and Jules Verne's Journey to the Center of the Earth. The setting is in a long lost land deep underground with its own Sun. Inside are dinosaurs, monstrous creatures, and other unearthly things.
Sannikov Land by Vladimir Obruchev
Similar to Plutonia in its Lost World aspect, Sannikov Land instead explores a long lost land in Russian and Siberian folklore called, you guessed it, Sannikov Land. It is North of the Northern most islands in the Siberian Arctic. The takeaway on this bit is that some believe this land to have actually existed and Obruchev was actually a geologist when he wasn't writing.
The Garin Death Ray by Aleksey Tolstoy
Not to be confused with Leo Tolstoy, author of War and Peace. Tolstoy borrowed the concept of a "death ray", popularized by H.G. Wells in War of the Worlds, and added an astounding level of technical detail to the idea. It is designed by an engineer named Garin who plans to use it to take over the world.
Selected Works by Grigory Adamov
Grigory Adamov's novels all share some common traits, so I'm going to throw them together. Conquerors of the Underground is a1937 novel, a highly technical, spy thriller. The Mystery of the Two Oceans, in 1939, was so technical that Adamov had collected hundreds of academic papers and articles to help him write it with expertise. The Ousting of the Ruler explored the idea of artificially warming the Arctic to un-icelock the Soviet coast. In his research, Adamov personally went to the Arctic Circle in the Soviet North.
#Late Soviet Fiction (Cold War Era)
We'll exclude Isaac Asimov because though from the Soviet Union, he did not live there most of his life.
Late Soviet Fiction, as a loophole in the censors, became very focused on ideas of utopia, dystopia, and ethics. Some utopian ideas were based on a future Communist paradise. Space operas remained fairly unpopular, though attempts were made. Post-Stalin fiction often looked far further into the distant future than Stalin-era and Pre-Stalin fiction. Here's a selection of famous fiction and descriptions of them.
Monday Begins on Saturday by Boris and Arkady Strugatsky
This 1965 novel is a satire of Soviet science. It is set in a closed-city in Karelia, where scientists research magic. You could try to compare this to Harry Potter, though I'm sure that would open up a can of worms for fans of this book and fans of Harry Potter. Some may argue that because this book uses magic it is actually a fantasy novel, but as I said before, Soviet censors did not much care for fantasy. As such, this book was covered with just enough science to get it categorized as science fiction.
Andromeda: A Space Age Tale by Ivan Yefremov
Andromeda, 1957, is one of those utopian stories I mentioned. It is set in a Communist utopia in the distant future. The main characters of the story fight aliens, but the primary focus of the novel is social commentary and cultural analysis of this society. This novel was considered a breakthrough in Soviet fiction due to it looking so far ahead into the future, where others had not yet done so.
Alisa's Adventures by Kir Bulychov
This is actually a collection of dozens of short stories and one of the few examples of a successful Soviet space opera. It was started in 1965 and is set in the late 21st Century. The main character is Alisa Selezneva, a teenage schoolgirl who's father is a scientist. Alisa has varying adventures through space and time travel. This series is a collection of children's stories.
Reading this, you may be reminded of the early episodes of Doctor Who which began in 1963, focused on an alien grandfather with a time-and-space machine, ferrying about his grand-daughter.
Roadside Picnic by Arkady and Boris Strugatsky
This 1972 novel is dystopian in nature, reminiscent of War of the Worlds or may even loosely remind some people of Fallout or District Nine. The premise is post-apocalyptic; aliens visited Earth in six "Visitation Zones". Within these zones are a number of supernatural events, artifacts, or phenomena.
The book was originally censored entirely and suffered serious delays. /u/OnStilts had mentioned the 1979 film Stalker; this movie is loosely based on this book and the screenplay was written by Arkady and Boris. The video game series S.T.A.L.K.E.R is even more loosely based on the book.
#Conclusion
I hope this answers your question. It is primarily literature, but I'm not much of a Soviet film buff. In summary, Soviet science fiction tended towards technical "hard fiction" and avoided "science fantasy" and "space opera". Some of the work was directly inspired by, highly reflected, or inspired in turn, similar works in the West.
Edits
I am prone to typographical mistakes.
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I cannot speak about rest of Easter Block, but in Soviet Union speculative fiction did exist, and it was large part of my childhood and young adulthood.
I. Aliens in the Emerald City
Let's start with fantasy. This genre was probably least in the good graces of the regime of all. For the most part it was permitted only for children consumption and mostly consisted of the retelling of the folk tales and classical stories, often with ideologically correct twist. Good example is "Jūratė ir Kastytis" (1959), animation based on Lithuanian mythological folk tale. Original story tells about fisherman Kastytis, who falls in love with sea godess Jūratė, her fiancée (god of thunder) get's jealous, smashes Jūratė's palace, lovers dies (in some versions Jūratė survives and remains chained to the ruins of her palace). In Soviet version, bout lovers survives, and Jūratė denounces her position as goddess to live as mortal fisherman wife. Then god of thunder strikes gain and kills her, prompting Kastytis to denounce gods.
After the Thaw, things become more relaxed and anvilicioussness decreased significantly, but not disappeared.
For example Nosov's "Know-nothing" trilogy. first book (1954) tells about childlike sort-of-hobits/smurfs and the silly shenanigans cosed by lovable rascal Neznayka ( Незнайка,meaning Know-nothing), second one (1958) have a magic wand as plot devise to transport protagonists in to the futuristic setting. And then there is a third one. Released in the 1966, Know-nothing visit the Moon, features extra-planetary exploits of our heroes were they have to encounter a capitalistic society.
Another popular series was Alexander Volkov's unlicensed sequels to the Frank Baum's The Wonderful Wizard of Oz. The first book (Урфин Джюс и его деревянные солдаты, Urfin Jos and his Wooden Soldiers. 1963) tells, about power hungry man who builds an army of wooden robots and almost succeeds in conquering the Wonderland. The final book- Secret of the abandoned planet ( Тайна заброшенного замка 1982) tells an amazing tale about how Emerald city had to fight of an alien invasion (Spoiler: They succeed because invaders were a slaver race and our heroes simply gave some magic potion to the interstellar Spartacus). I cannot comment on the other 3 books, as I haven't read them.
That said, I must stress that Volkov and Nosov were good writers and- despite insane plots- they produced a some good children fantasy (Know-nothing visit the Moon is worst of the lot, utter trash but it's interesting to read trash nonetheless).
As for the more adult-oriented fantasy, regime was far less forgiving. Even Alexander Grin was pushed aside during the reign of socrealism and only during the Thaw he was remembered. And his only crime was to set his stories in the fictional land. Otherwise Grin's stories a rather run of the mill late 19th- early 20th century. His biggest crime (aside from fictional land) was lack of politics- he wrote adventure and love stories without much regard for revolutionary ideas. His most popular work Crimson sails ( Алые паруса) is a sappy star-crossed love story, that features a guy who claims to be wizard.
Even Master and Margarita was problematic in the regime eyes (not only because of the Devil, but Bulgakov's choise of the dramatis personæ did not helped).
In short, fantasy elements could exist freely in Soviet Union only in children stories and anti-religious pamphlets. Otherwise they should be either an obvious allegory or a classical work predating USSR.
Just to give an example, move based on the Gogol's story Viy (Вий, 1967) was an sensation among the populous of USSR, just because it was a mystical horror story about possession. Nothing like that was made in USSR before and it remained anomaly for quite some time.
Monday Begins on Saturday (Понедельник начинается в субботу) and tale of Troyka (Сказка о Тройке 1968) by the Strugatsky brothers was pretty much the on the limit of how much fantasy elements one could put in the story and still get published.
II. Atlanteans on Mars
Now on science fiction. This get much more freedom, albeit still with limits. A good example of how science fiction was subjugated to the needs of the regime is movie Cosmic journey ( Космический рейс 1936). Now celebrated as getting space travel surprisingly right (Tsiolkovsky himself helped with production), movie was only produced because Komsomol wanted to increase interest in space among youth. And was immediately shelved after initial screening, because censors decided that this low-gravity scene are undignified and harms Soviet image.
The best remembered example of the early Soviet science fiction is- without contest- is Aelita ( Аэли́та 1923) by Alexey Tolstoy. Novel is about a Soviet engineer Los who builds a spaceship, hires Civil War veteran Gusev as additional muscles and flies to the Mars, where he falls in love with the daughter of the Mars leading politician. Most of the novel deals with this plot-line (stopping shortly to tell the history of Atlantis), while Gusev organize a revolution against Aelita's father regime. That fails. Hard. Two earthlings just manages to escape, while Aelita gets captured and imprisoned by reactionaries. Novel ends with Los receiving radio message from Aelita.
It seems that in west Aelita is better known because of the movie adaptation, but I find book to be better.
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However, to speak about early Soviet science fiction is to speak about Alexander Belyaev. Despite occasional flirt with fantasy elements (Ariel (Ариэль) 1941), he stands strongly in science fiction camp, and produced a number of works that today would be classified as soft fiction. Best remembered (mostly due to 1962 movie) is Human-amphibia ( Человеκ-aмфибия 1928) about a professor who saved his son by transplanting shark gills, thus enabling him to life underwater (transplantation is recurring in the theme Belyaev's works).
However most of the works in the genre were much more ideologically oriented. The most of the pre-Thaw science fiction shares the same structure: setting is contemporary or near future, central plot device is a scientific discovery of some sorts and central conflict is some evil capitalist/imperialist attempting to use said discovery for his own gains.
While this structure also fits Aelita and Human-amphibia, this two works are, lets say more centered around characters rather than conflict with class enemies.
Belyaev's Air seller ( Продавец воздуха 1928) is however much more fitting the bill for the "typical soviet science fiction story". The story deals with evil capitalist who build an air-sucking devise in order to take entirety of the Earths air in to his possession and then profit from selling it. Author strongly implies that main antagonist Beiley is just a front-man for much larger western capitalist conspiracy and story ends with Red Army storming Beiley's facility (Evil overlord tip #7783: Do not build huge air-sucking devises in Yakutia).
Aside from ideological angle, it seems that most Soviet writters were stuck in the Jules Verne tradition. The only noteworthy deviation is Bulgakov's Dog's Heart ( Собачье сердце 1925) (If Bulgakov can be called a fantasy or science fiction writer at all).
Forgotten treasure of the period is Yevgeny Zamyatin's We ( Мы). The novel is one of the first dystopia's and certainly the first written with the experience of living in totalitarian regime.
Up until the Thaw there was no significant deviations from this model, however if you have a taste for Vernian style pulp fiction, Soviet Union have a lot to offer.
The milestone of the Soviet science fiction is Ivan Yefremov's Andromeda Nebula (Туманность Андромеды 1957, made in to movie 10 years later). The previously discussed structure of the Soviet science fiction was not as much rejected by Yefremov, but outright ignored- the setting is far future, the antagonist- if any- is a distance of space and only plot, is a space expedition and Strugatsky's cite it is as a major inspiration. It is also... bad. Like, really bad. Instead of characters we have one-dimensional inhabitants of the perfect communist future. Action and adventure oriented stories can get away with shallow characters, but Andromeda does not have much then it comes to action or plot either. The book importance lies in the context of publication- it broke established mold, but as a piece of literature on itself. it has nothing. If you want to read Yefremov In the Land of Oikomena (На краю Ойкумены 1946) is much more better work of his.
And so began the era of Strugatsky brothers. It is disservice for other writers, but they do cast shadow over the entire genre. Compared to the previous era, the new works are staggeringly dark (Roadside picnic Пикник на обочине 1979) made in to movie Stalker. While Stalker is indeed considered one of the best science fiction movies in the west in former Soviet Union Solaris (Солярис 1972) (directed by the same Tarkovsky) is better remembered. Just to illustrated how situation changed from Vernian tech-happy stile, one of the last science fiction movies produced in USSR Witches cave (Подземелье ведьм 1989) deals with danger of playing God. It's by no means the best take on the topic (low production values does not help) but the screenplay was written by Kir Bulychev, best known for shiny-happy Alisa Seleznyova series (a golden standart for children oriented science fiction).
Another theme was more or less open critique (more often less) of the regime or just simple attempt to bite the censorship, with Strugacky's again paving the way with Monday Begins on Saturday (Понедельник начинается в субботу 1965).
Well, I hope someone can take it form here. While I love Strugatsky brothers to no end, they are pretty much 95% of the post-Yefremov science fiction I consumed.
Oh for the end note, back 1985 Soviets produced live-action Tolkien's Hobbit. Just so you know.
Hi, I just ran into this. By the way, there's a collection of Soviet fantasy and sci-fi films that circulated a few years ago: definitely check it out if it comes to your area. Anyway, here are a few earlier posts that you may enjoy:
What was genre fiction like in the Soviet Union? featuring /u/Noble_Devil_Boruta and /u/Akerlof
cont...