The Homeric myths and stories were a definitively oral tradition, passed in by word of mouth. But our text translations today go back at least as far as medieval Europe. Where do our current ‘definitive’ Iliad and Odyssey come from, and when was it penned?

Okay, well, this is like, a crazy complex question with huge amount of scholarly debate, although a lot of fun. Even the TL;DR is long:

The Iliad and Odyssey have been been circulating as complete editions in written form from at least from the 6th century BC, but an idea of a 'true' and 'authentic' Homer probably goes earlier than that. The basic storyline, themes and style were always there, but the poems continued to exist in somewhat varied textual versions, until an important watershed moment in the 2nd century BC Alexandria, where a hub of grammarians sought to remove all 'unauthentic' variants and errors and collate one 'original' version of Homer. The result of their work became the standard version of Homer, sometimes called the 'Aristarchus' version', that was in popular circulation for the rest of antiquity, and we have substantial manuscripts of large chunks from every century up to the 7th century; the texts were, however, always in flux - addition of a line here and there etc. - but the variants gradually become more and more sparse with time. The earliest complete texts only date to the 9th-10th century CE. These are more or less the versions of Homer we use today, although scholars continue to debate what exactly the 'true' and 'authentic' Homeric versions (often titled as the 'vulgate' Homer) were, through all this fog and chaos of c. 2700 years of oral and written transmission; each new edition of the Greek texts are different and scholars can still majorly disagree on many points, so there isn't really such a thing as the one 'definitive' Homer.

Okay, and the long version. Already in antiquity the 'correct' version of Homer's epic was a self-consciously contested issue, although it remains an open question whether there truly ever was one complete poem created by a person called Homer; ancients believed so, certainly. Later commentators, such as Strabo, Cicero, Plutarch, and some scholiasts, believed that Peisistratos, tyrant of Athens (c. 561-527 BCE), realised that there were 'confused' and wrong versions of the true, pure Homer in circulation, and laid out a definitive 'master Homer'; this is sometimes known as the 'Peisistrean edition' in scholarship, but whether one ever really was made by a histrocial Peisistratos is impossible to say. Some sort of full versions of Homer definitely existed in written form in classical Athens. Notably, Plato's quotations of Homer differ quite a lot in quality from Aristotle's quotations of Homer; Plato's are considered as 'pure' and 'correct' in that they correspond almost perfectly with what was later coined as the authentic Homer (titled in the scholarship often as the 'Vulgate' Homer to differentiate from 'unauthentic' variants, although an extremely elusive and contested concept), whereas Aristotle's variations are considered as errors or unauthentic additional lines, which shows at least that Plato and his student Aristotle had differing editions.

We can best start studying the formation of the 'Vulgate' Homer once we start having something that isn't just out-of-context chance quotations, but actual editions of the text. From c. 3rd century BC onwards, we have scattered papyrus versions of Homer - these are never the whole poems, only larger sections. About forty such papyrus 'manuscripts' dating from before c. 150 BC were known to Michael Haslam in 1997, who I'm using as my source; I don't know if any have emerged since then. These do, however, show quite remarkable level of variability, meaning, verses have alternative spellings; differing choices of words; sometimes extra words, sometimes less; sometimes a full verse or half of a verse is found in a 'wrong' place; and there are, indeed, some handful of verses that are completely unknown in the later tradition.

A real watershed moment to Homeric transmission is the c. 2nd century BC, when the science of grammar begun to take shape in Alexandria. A hub of scholars started to take really seriously the idea that there is one 'pure' and 'correct' version of Greek, and the original Homer was held as a semi-divine paragon of the perfect Greek. Thus, the Alexandrian scholars, discontent with the variability and quality of contemporary versions, 'corrupted' by centuries of transmission, set out to 'restore' Homer's original poem. We know quite a lot of this process because of the tradition of ancient textual commentaries ('scholia') who tended to, similar to modern academic traditions, note down changes and opinions of 'previous editors'. The Wikipedia article actually does a pretty good job in summarising what this exactly meant, if somebody wants to read more. This went beyond just grammatical, metrical or lexical considerations. Any inconsistencies in logic or style were considered as corruptions, and thus they were fixed. Too much repetition of the same word or line too close to each other, fixed. This is, actually, what modern scholars still do when reconstructing an ancient text from differing manuscripts. But, the Alexandrians went sometimes a bit far in comparison to modern scholarship. A word too weird, fixed. A line of too 'poor' poetic quality, fixed (the divine Homer could never write something mediocre!). Questionable morals portrayed in the text? Homer wouldn't do that, fixed! There are some indications in the later commentaries that the Alexandrians might have rejected whole passages as 'un-Homeric', which they indeed might have been (these guys really were dedicated, thorough and knew their Greek), but we of course can't ever find out if we would agree. The most notable of these Alexandrian grammarians were Zenodotus, who probably established the tradition of dividing both Iliad and Odyssey into 12 books each and put the narrative in the order we know today; and Aristarchus of Samothrace, who created an overhauling, major critical editions of both Iliad and Odyssey.

These editions of the Alexandrian school, the 'Aristarchus' version' became more or less the standard that was popularly circulated throughout antiquity. However, the tradition continued to be joyfully in flux, open to further debate and disagreement, as each new commentator had their own idea of what the most Homeric Homer was. We have editions of major chunks of Homer from practically every century down to 7th century CE, and no two versions are exactly the same. However, variants become more and more minor and less and less sparse through time; the first fully preserved versions of the Iliad and the Odyssey date to the 9th and 10th centuries. This is basically the tradition of transmission from which we get our Iliad and Odyssey today.

However, scholars continue to heavily disagree what style, wording, ordering is truly 'the most authentic Homer', the 'vulgate' version of the texts. Some believe that the pre-Aristarchus version was closer to the original Homer, some believe that the creation of Aristarchus et co. was as comfortably close to the true master Homer as we will ever get to (evidenced by the Plato quotations which well correspond to it, perhaps?). Some scholars, in turn, believe that the whole idea of a 'vulgate' Homer is a wild goose chase and we should just be comfortable to think of it as a multi-text tradition. In any case, each new edition of the Greek Homer still has to contend with this massively long and confused tradition of centuries of transmission, compare different manuscript versions, consider how seriously old papyri versions should be taken and so forth. As a consequence, we still have no 'definitive' Homer, and it's still an evolving text (though, admittedly, ignoring the more radical calls to go back to the pre-2nd c. BC versions - we are mainly talking about minor variants that are mainly of interest to serious scholars of Greek language and literature, not something that would get reflected in English translations).