Han shot first. It was only after Pro-Han revisionists examined the evidence that opinions shifted to Greedo.
There is no scholarly debate on this largely due to the eyewitness account of Wuher, the bartender of the Mos Eisley Cantina where this event occurred. Wuher states that Greedo threatened Solo and Solo shot him from under the table. He was also quite upset that Solo only gave him a coin "for the mess" because most transactions are done using credit chips and a coin was almost pointless. He didn't press the issue because Solo seemed like a bit of a wild card and one of Solo's associates had cut the arm off one of his regulars, Ponda Baba, and his staff was "already busy cleaning blood from the floor so what was the harm in cleaning a little more?"
When asked why he would not simply use a maintenance droid, Wuher becomes confrontational and kicks people out. He does admit that there is a holodisk recording of the event, but a powerful denier, strong in the Dark Side, has poorly edited and released copies of the altered version to make it seem as if Greedo shot first. This is an obvious fake as Solo's head moves unnaturally and there is no scorch mark on the wall where Greedo's shot allegedly hits. This edit is the source of the confusion.