The closest equivalent I can think of are the Mahabharata and Ramayana in India. Both these two poems and the Odyssey and Iliad have certain elements in common because of their mutual descent from Proto-Indo-European practices. The bit about reverence and almost scriptural role that u/White___Velvet mentioned is also true about the Mahabharata and Ramayana, the famous Bhagavad Gita is a part of the Mahabharata, for example.

The Odyssey and the Iliad are epic poems, probably made to be recited rather than read and possibly handed down orally before being written down. Thus, if the question is what I understand it to be, I would say that pretty much all cultures have (literary or orally) equally important epics and myths.

One very close equivalent is the vedic Mahābhārata, which stem from approximately the same time period as the Homeric poems (about 400BC), but (like the Homeric) may have even older roots. AT >200000 verse lines it dwarfs even the Iliad and Odyssey, yet it is still performed orally by memory today.

The Gilgamesh epos of the ancient Sumerians is another obvious example, having survived in many forms from a long time period, and shaped flood myths around the whole Middle East.

I think you can argue that some ancient Egyptian traditions like the Story of Sinhue from the early 20th century BC also may be fairly analogous.

The Epic of King Gesar is younger, 12th century AD, but may be considered as a Mongolian counterpart.

The Norse equivalent may be the two Eddas -although also written down in the 12-13th century, they record and summarize a much larger mythology and history that reach back into the "Viking age" and earlier, and certainly have been a basis for modern Scandinavian self-understanding.

In short, the question is a bit wooly, there are so many works and traditions that may be considered depending on how you define your question.

So this is going to hang almost entirely on what you mean be "equivalent to" the Odyssey and Iliad.

Do you mean "Ancient, epic poem held in high regard by both us and the ancients"? If so, then things like Virgil's Aeneid would certainly qualify. The Aeneid even explicitly links the founding of Rome to the events of Trojan War as relayed in Illiad. The Epic of Gilgamesh also looks like a likely candidate. Or what about something like Beowulf? It is arguably "ancient" and held in pretty high regard, at least contemporarily. But where is the line for "ancient"? Does the Song of Roland count? The Cid? Paradise Lost? All these poems are treasured epics from past ages, after all.

Or maybe the thought is that the Odyssey and Iliad occupied a special, almost scriptural role for the Greeks. They certainly are generally mentioned with reverence by the Greeks themselves. Maybe you think Beowulf fails in this regard, that it wasn't enough of a universal cultural touchstone to truly count as "equivalent" to the Odyssey and Iliad. Maybe a better candidate might be the Quran, a long book of poetry that purports to relate history, morals, and something of the underlying truth of existence.

In any event, I think the Aeneid should count, given its age, esteem, and status as a cultural touchstone in post-Augustan Rome. As for the others, again I think it depends on what you take an "equivalent" work to be.

The Old Testament!

I think we should count that? It's definitely an Epic and it's effects on ancient (and modern) cultures are everywhere.

The Old Testament began as an oral tradition amongst the Ancient Israelites and served as the narrative of their history. It's obviously of a much broader scope than The Odyssey and The Iliad, but I don't think that can detract from saying that it's a literary equivalent to those two epics. It's tied more to the Israelite and Jewish identity (not to mention every form of Christianity) than The Odyssey and The Iliad are to the Greek/Western identity.

Luo Guanzhong's Romance of the Three Kingdoms is regarded as an epic classic and it is probably most influential literary work of China.

An another one is Popol Vuh, a cultural narrative that recounts the mythology and history of the Mayas.

In what is now India, the Sanskrit epics Ramayana and Mahabharata would certainly fit the bill, and predate or are at least contemporary with Homer.

The German Nibelungenlied survives in manuscript form from the 1300s and is considered the national epic of Germany.

The Finnish Kalevala only dates from the 1830s in its complete form, but the poems and legends that compose it are much older.

The Icelandic Eddas date from the 1200s in written form, and again were based on much older tales and myths.

Apart from the Mahabharat and Ramayan of India, the Jataka Tales which chronicle Buddha's (founder of Buddhism) previous births may also fit the bill (i.e the Question asked here). I say "may" because this is a collection of tales, and not a continuous single tale and hence not grandiose or epic.

However, culturally, the tales are important in Buddhism. They contain 547 poems and are among the earliest forms of Buddhist literature. Buddhist monuments such as the Borobudur contain scenes and quotes from this piece of literature. Today the tales are part of the folk tales of India and many of them are performed on the stage and in films in Buddhist countries.

If the Jataka Tales fit the bill, then another contender would be the Panchatantra, another collection of tales. These however are non religious and are attributed to a sage who used them as a medium to instruct Princes in Politics and leadership. These tales also form a part of the folk tales of India and are one of the most translated books in history.

In Ireland, the equivalent is the Ulster Cycle, particularly the Táin bó Cuailnge, or Cattle Raid of Cooley, which tells the story of a war fought over an especially impressive brown bull. It was composed in the 7th century and contains a lot of the foundational myths and heroes of Ireland.

I misread the title to be a question about what the equivalents of The Iliad and The Oddyssey were to these ancient cultures in their times. Which I'd really like to know actually.

You guys are brilliant and helpful and simply incredible. Thank you for giving me reading material for a year.

Not as old, but in Japan, the Kojiki (8th Century).