Poor Harry Potter is stuck spending another summer with his odious relatives, the Dursleys, who really should've had Child Protective Services called on them years ago. After putting up with lots of rotten treatment, Harry finally loses his temper and blows up his Aunt Marge (like a balloon, not like a bomb). So Harry hightails it out of Number 4 Privet Drive and gets picked up by the Knight Bus, which takes him to Diagon Alley and thus back to the wizarding world.
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Harry isn't punished for his "crime" (he's not allowed to practice magic outside of school) and soon finds out why: turns out, a dangerous prisoner, Sirius Black, has escaped from Azkaban and is hunting down Harry. Given how many times he's faced off with homicidal maniacs, Harry isn't all that concerned about this development. The real problem is that he doesn't have permission to visit Hogsmeade, the wizarding village near Hogwarts, this school year. Now this is a serious bummer.
Harry reunites with his two BFFs, Ron and Hermione, and the three head back to Hogwarts with something new in tow: Hermione's adopted cat, Crookshanks (who is rather hideous and super smart). Crookshanks, of course, has it in for Ron's pathetic rat, Scabbers, which leads to a long series of arguments between Ron and Hermione. So what else is new?
On the way back to school, some Dementors, the scary guards of Azkaban, enter the train to look for Sirius Black. Being around the creepy Dementors makes Harry hear his parents' death at Voldemort's hand, and often makes him faint (not something that most teenaged boys hope for).
At Hogwarts, changes are afoot. Remus Lupin arrives as the awesome new Defense Against the Dark Arts teacher, and Hagrid is the new Care of Magical Creatures teacher. The year progresses and Harry is busy kicking butt in Quidditch, Hermione is swamped with her insane course load, Hagrid is eagerly (with Hermione's help) trying to prove that his hippogriff isn't dangerous before a tribunal, and Ron is consumed with worrying about Scabbers. Meanwhile, everyone is preoccupied with fear of Sirius Black, who breaks into Hogwarts twice during the school year.
Harry devotes most of his time to three things: Quidditch (which pays off since he and his team win the Quidditch Cup, finally), sneaking into Hogsmeade (courtesy of a magical map provided by Fred and George), and working with Professor Lupin on casting the Patronus spell, a defense against Dementors.
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Midway through the year, the boys stop speaking to Hermione after her cat presumably kills Scabbers and after she turns Harry's anonymously gifted Firebolt broomstick to Professor McGonagall for inspection. The trio eventually reconcile in time to help Hagrid with his eleventh-hour appeal to save Buckbeak, his hippogriff. Harry gets a cryptic spoiler alert about the book's climactic moments from the Divination teacher, Professor Trelawney, who predicts that Voldemort's servant will return to his master. Yikes.
Hagrid loses the appeal. On the day of Buckbeak's execution, the Harry, Ron, and Hermione find Scabbers alive and well in Hagrid's cabin. Things get nuts after this. Scabbers runs off, a giant black dog attacks and drags Ron and Scabbers into the creepy Shrieking Shack, Ron's leg gets broken, and Harry and Hermione follow. Soon Lupin and Snape show up and we learn the shocking truths: Lupin is a werewolf, Sirius Black is an Animagus (meaning he turns into the giant black dog at will) and Scabbers the rat is really Peter Pettigrew, Voldemort's servant and the real criminal (Sirius was framed), hiding in his Animagus form. Crazy. As it turns out, Sirius Black is a good guy and Harry's godfather.
Pettigrew escapes and Sirius is captured by the Dementors. Before we can get too depressed, though, Dumbledore arrives and instructs Hermione to use her Time-Turner to fix everything. So Harry and Hermione travel back in time to that afternoon and manage to save both Buckbeak and Sirius. Harry saves Sirius by casting a super powerful Patronus against the creepy Dementors (he didn't even faint!).
The school year ends and everyone heads home for the summer. Lupin is forced to resign (because he's a werewolf); Sirius, who is now in hiding, writes Harry a letter; and the BFF trio parts ways for yet another summer.