The Peter-Weyl theorem is one of the results that made me decide to study representation theory. In a few words it tells you how to describe the space $L^2(G)$ in terms of the representation theory of a compact group $G$. The idea of this talk is to informally develop enough theory to state and understand this theorem and some of its consequences, and in this way motivate the study of Lie groups and their representations. No previews knowledge of the subject is assumed.