Geometry is nothing but an expression of a symmetry group. Representation theory is the study of how such an abstract group appears in different avatars as symmetries of geometries over number fields or more general fields of scalars. It is a technique for analyzing abstract groups in terms of groups of linear transformations. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication. Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. Representing Finite Groups provides comprehensive overview on the basics of representation theory for finite groups from the point of view of semisimple algebras and modules over them. This book will be of valuable tool for professors, practitioners, and researchers in mathematics and mathematical physics.