The Laplace operator is the infinitesimal generator of Brownian motion with a real state space.  The Vladimirov operator, a $p$-adic analogue of the Laplace operator, similarly gives rise to Brownian motion with a $p$-adic state space.  This talk aims to introduce the concept of a $p$-adic Brownian motion and demonstrate a further similarity with its real analogue: $p$-adic Brownian motion is a scaling limit of a discrete-time random walk on a discrete group.  Attendees need not have prior knowledge of $p$-adic analysis, as the talk will provide a brief review of necessary background information.