The simple idea of representing algebraic objects as linear operators on a vector space has applications to many branches of mathematics, including algebra, number theory, geometry, harmonic analysis, and physics. For us, the story will begin with some elementary representation theory of finite groups and the role of the regular representation. This representation is on the group algebra, which is an algebra of functions under convolution. Since this algebra encapsulates many of the important aspects of the representation theory of the given group, it may be used to characterize certain properties and simplify certain questions. This notion carries over to more general types of groups, some examples of which will be discussed. In particular, the algebra of interest for a compact group will be that of square-integrable functions; for a p-adic group, the relevant algebra is the affine Hecke algebra.