Alfonso Cuaron was as close as the Harry Potter franchise came to a visionary director. I can't imagine why they didn't keep him on for the rest of the movies. Cuaron swathes the film in a new, richly gothic atmosphere & finally gets more gritty performances out of the child cast. In the adult cast, Gary Oldman as the titular escaped convict Sirius Black & David Thewlis as the new Defense against the Dark Arts professor, are excellent as Harry's damaged, morally ambiguous mentors.
Cuaron's directing, Michael Seresin's cinematography & John Williams' Danny Elfman-esque (I could probably get shot for saying that) score lend a sense of weight & momentum right from the outset. Cuaron's Dementors are thrilling creations, far superior to the ones in Order of the Phoenix. Cuaron nails his tone, & sets the scene for the finale, with an early shot of a bluebird leading the camera around early morning Hogwarts scenery before being chomped by the Whomping Willow.
Harry's breakaway flight on Buckbeak the Hippogriff is one of the series' most stirring & poetic scenes in the series; a rare happy moment for Harry & a triumph of visual effects. The sets are exceptional, even by Harry Potter standards, Hogwarts and surroundings have never looked more stylishly gloomy, the werewolves are cool & tragic, the magic bus is a blast and the nifty time travel plot is smartly executed. Prisoner of Azkaban has no detractors. Easily one of the best in the series.
Interesting to note that Mexican born Cuaron's presence ushers in the first non-white Hogwarts students, even if they are confined to the background.
Oscar noms: Original Score (John Williams), Visual Effects.