Just for youngsters, i.e. anyone under thirty-five, here is a clip from Star Wars Episode IV: A New Hope, showing that Han shot first, and indeed was the only one to shoot, because he hit Greedo, obviously, and there was no return fire. (Thanks to uploader Daniel M Kobayashi.)
Whenever this one comes up, there's always a few people who don't care who shot first. They don't care that Greedo (in the Special Editions) missed at point blank range, and they don't care that Han waited for him to shoot first, a non-survival trait that really seems out of character for him as we first meet him.
As far as I'm concerned, Han not only did, but must have, shot first because he thinks only of himself - which he continues to do for some time. That's why he's an interesting character. When he starts thinking of others, and even helping other people for free, that's when you know the character has grown.