'Gilgamesh' is the Akkadian version of the name for a king of Uruk; in Sumerian sources he is sometimes called 'Bilgames'. Our stories of him probably go back to various Sumerian poems, perhaps originally from the Third Dynasty of Ur, in the 21st century BC, of which a few late recensions here and there survive, but the earliest extant text of the tale of Gilgamesh is known as the Old Babylonian version, which survives in a few tablets, and is from the 18th century BC. A Hittite version, rather different in parts from the main stream, was worked up a few centuries afterward. A newer version, the Standard Babylonian version, was put together in perhaps the 11th or 12th century BC, and survives in fragments that have come to us through the rediscovery of the Library of Ashurbanipal in Nineveh in the 1850s. There are many other fragments, however, across this expanse of time. The destruction of the Assyrian Empire seems, however, to have mostly swallowed the Gilgamesh tale, as it did much more of the cuneiform literature; its resurrection is one of the great achievements of modern archeology.
I will be reading The Epic of Gilgamesh in the Penguin Classics edition translated and edited by N. K. Sandars. Like all other versions, it is a reconstruction, following the Assyrian collation in great measure, but drawing on other sources, such as the Old Babylonian; when Sandars first worked up a version in the 1950s she drew more heavily on the scattered Sumerian fragments than was common at the time.