There are many obvious similarities between the flood story in the Epic of Gilgamesh, and the story of Noah's flood. This of course brings up the question of whether one was inspired by the other, particularly the story of Noah being inspired by Gilgamesh, which is the older flood account.
So to start, let's look at some of the similarities:
1. God (Genesis) and the Anunnaki (Gilgamesh and related stories, i.e: Atra-hasis) decide man must be wiped out by a flood.
2. An individual (Noah and Utnapishtim) deemed worthy to live by God or one of the Anunnaki is given warning of the eminent deluge and ordered to build a boat to save their family and chosen animals.
3. Both Noah and Utnapishtim send out birds to see if the flooding is going away.
4. Each makes sacrifices to God or gods once settled on dry land.
Those are just the most similar connections, however here is a link that gives a bit more in depth similarity studies side by side.
As I said earlier, the question is, was one inspired by the other, and if so what does that mean. Clearly there are many who take the Bible literally and feel that the story of Gilgamesh just lends more truth to the story of Noah's Flood. And while it does indeed do that, it is really a double edged sword. The fact that the Gilgamesh account was recorded before the book of Genesis was written tends to lead many to the assumption that perhaps the story of Noah was inspired by that of Utnapishtim. Unfortunately there is no proof either way, excepting a small bit of evidence that might prove the existence of a couple kings mentioned in the Gilgamesh epic which wouldn't do much to support the Noah believers.
Also if Noah's story was inspired by the tale of Utnapishtim in Gilgamesh, who is to say that perhaps that story wasn't inspired by an earlier tale as well? Most of our knowledge of the early civilizations of Mesopotamia is fragmented at best and much more is probably waiting to be found there. Who knows what the future holds.
Sources and further reading:
The flood story from the epic
Gilgamesh Flood Myth - Wikipedia
Is the Biblical Account of the Flood Modified from the Epic of Gilgamesh?