As you already know, Giovanni is a major figure in the whole saga involving Mewtwo. His mother, who the fans affectionatly call "Madame Boss" started the project by looking for Mew. It was under Giovanni's command that Team Rocket found a folssilized eyelash of Mew, and the cloning began. Gio wanted to make improvements on the original mew, thus MewTwo. Not only is Gio mentioned through the whole ordeal, but he shows up.
In the first Pokemon movie, Mewtwo Strikes Back, Gio shows up in the prolouge. He is in shadow and lures Mewtwo into working for him after he blows the lab of scientists. Then Gio makes the mistake of telling Mewtwo pretty much "I'm using you. Ha ha" ok, so he didn't use those exact words, but shoulda. Same thing woulda happened. M2 gets pissed off, blows the HQ (almost blows up Gio, but he lives!) and blah blah bblah ya know the rest.
In Mewtwo Lives (which I DLed here) is a sequel to the movie. Mewtwo has been living on this island thingie with this water that heals stuff. *shrugs* to me, I thought it made everyone drunk. They sure acted like it. Anyway, there's this girl who's like Gio's right-hand Rocket, any way, Gio sees Mewtwo while they're looking at a pic of the island and then...stuff happens *shrugs* I don't speak Japanese. Anyway, Gio captures mewtwo and TRs working on the water stuff and lotsa things blow up. Heh! lovely summary huh?