What is certain is that the ancient Greek scholars and commentators were convinced that Homer was real and lived in the 9th or 8th century B.C. Modern scholars generally agree that the Iliad was composed around 725 B.C. (the earliest written versions we have are hundreds of years later than that, so there’s plenty of room for conjecture). But though we don’t have the earliest texts, the ancient Greeks did, and Homer was written about, discussed, and analyzed throughout the classical Greek period.
One of the key controversies among Homeric critics is whether Homer composed his poems orally or whether he actually wrote them down. We do know that Homer’s poems were recited in later days, at festivals and ceremonial occasions, by professional singers called rhapsodes, who beat out the measure with rhythm staffs. (There is a similar poet/singer in the Odyssey who sings a poem about the Trojan War. He is an old man, and blind; that may be the source behind the legend that Homer himself was blind.) Whether or not Homer actually wrote down his poems, it now seems certain that the Iliad and the Odyssey are part of an ancient literary tradition of oral composition. The stories on which they are based had probably been sung aloud for hundreds of years, and recited and memorized by one generation of poets after another before Homer took them in hand. After all, in Homer’s time, writing was used mostly for inventories and business transactions. Recitation was the accepted means of relating myth and history.