Moonshine is a mysterious set of relationships between different fields of mathematics -- number theory, representation theory, and (most recently) algebraic geometry -- explained by their connections to certain special solutions of string theory. In this introductory talk we explain the basic objects involved as well as the original Monstrous moonshine conjectures (now proven). We also sketch recent extensions to more physically interesting solutions of string theory, such as compactifications on K3 surfaces and Calabi-Yau manifolds. The talk is designed to be accessible to a typical high energy theory audience.