The paradox of the falling a cat is a famous example in geometric mechanics. Simply stated, a cat in free fall can execute a 180 degree turn of its body, even though has zero angular momentum throughout the entire maneuver. In this talk I will discuss how this seeming paradoxical behavior can be explained through differential geometry and holonomy, which in turn can offer insights into the behavior of other mechanical systems. This talk is meant for a general audience and no knowledge of differential geometry will be assumed.