Not to be confused with certain small birds or equestrian harnesses, a martingale is a concept in probability with interesting applications to PDEs, functional analysis, and mathematical finance. We'll begin by motivating the idea of a martingale as a fair game, and from there give a neat characterization of a zany process called Brownian motion with an intimate connection to the heat equation. We'll then discuss some results concerning martingales in the areas of Hardy spaces and asset pricing.