Representation theory of groups is the study of groups (of all sorts: finite, infinite, algebraic, topological, etc.) acting on vector spaces (of all sorts: finite-dimensional, infinite-dimensional, topological, Hilbert spaces, etc.). It is an important area of study in its own right, and also finds many applications throughout mathematics. Loosely, one might say that the utilization of symmetries, and the linearization of problems, are two of the fundamental tools of mathematics. Combining them, one is directly lead to problems involving representations of groups on vector spaces.