This question is inspired by these Soviet Animations of Ray Badbury Stories, which was posted in /r/literature.
The US and Japan are the countries I know of that are most famous for animation. Before seeing the above videos I had no awareness of animated films in the Soviet Union. So, how did Soviet animation compare to work in the US and Japan in terms of artistic style, audience reception, popularity, and production?
Soviet animation is a fascinating subject, and one that rarely gets talked about. Unlike Japanese and European animation which was heavily influenced by American companies like Disney, Fleischer, and MGM among others. I talk a little bit about it here, Soviet animation was largely unaffected by western influence.
This lead to many stories and styles completely separate from western thought, including an animation style known as photographica which to this day has never been used outside of the Soviet Union. This technique is most famously used in the short film Polygon: The Firing Range. The result is unlike any western animated film ever developed.
A lot of people picture Soviet animation as being completely draconian, but it was actually one few professions that allowed a certain degree of creative freedom, allowing for a wide degree of audiences and subject matters. There's the humorous Последний бой м/ф (The Last Fight), the 2 part science fiction/horror АМБА (AMBA), and even a [Russian version of Winnie the Pooh](http://en.wikipedia.org/wiki/Winnie-the-Pooh_(film).
Popularity wise, most Russian animation is virtually unknown outside of it's homeland, but it's done very well within its own country.
Some names to know in Russian animation are; Soyuzmultfilm the largest, one of the oldest and most well established soviet animation company, and producer of the aforementioned Winnie the Pooh movies, and the most likely to be known outside of Russia. Kievnauchfilm one of the other major film studios, after the fall of the USSR, its film and animation departments were split into the National Cinematheque of Ukraine, and Animafilm respectively. Belarusfilm which often worked in conjunction with Soyuzmultfilm, produced several stop motion animated films.
Melnitsa Animation Studio a modern animation studio that has been gaining popularity of late. And finally, Digital Light Studio another modern russian studio that is known for 3D animated short films like Steam Speed, and S.W.A.T., so far all of their work can be found on their youtube page here.