Why do cats (usually) land on their feet? In the six-minute video below, missile engineer and last-name-avoider Destin explains...and includes some real stunt-cat dropping photographed with a high-speed camera. He also has archival footage of Air Force researchers dropping cats on parabolic flights, which helps explain the physics behind the phenomenon. As far as I can tell, no cats were harmed in the making of this video -- though they seemed, at the very least, displeased.
For some deeper math, check out the paper Gauge Theory of the Falling Cat (PDF) by Richard Montgomery. Warning: it involves differential equations.