This talk will describe the Laplacian on the Berkovich projective line and will use it to construct the "canonical measure" associated to a rational function of degree at least 2, which is analogous to the classical invariant measure constructed by Brolin, Lyubich, and Freire-Lopes-Man\'e. It will then discuss equidistribution theorems relative to this measure, and use them to derive structural information about the Berkovich Julia set.