When is it believed the Iliad and Odyssey were actually composed?
The Odyssey and the Iliad, both credited to Homer (though no one really knows much about him) are seen as the foundations of Western literature. Both works remain influential down to the present day.
The dates when these epics were composed is not clearly known. Scholars believe that the Iliad is older than the Odyssey and they believe that both works were compsed sometime in the 8th century BC (meaning the 700s BC).
Because there is no way of knowing for sure when the epics were composed, scholars have had to do such things as analyzing the fighting techniques portrayed in the story and trying to figure out from them when the poems were composed.
See the link for more about how they figured this out.