Group Representation Theory has many uses in Physics and Chemistry, representations of the symmetric group being the most widely used. This book introduces Group Representation Theory and discusses various methods in which to calculate representations. The first chapter introduces the subject creating a basis in general terms. The remainder of the book will focus mostly on the symmetric group. The second chapter derives a method to calculate representations of the symmetric group called Young''s natural representation which utilizes the fact that there is a one-to-one correspondence between Specht modules and the irreducible Sn -modules. In the third chapter the complex group algebra is decomposed demonstrating another method for calculating representations of the complex group algebra. These representations are equivalent to the corresponding group representations. In the last chapter multiple methods are discussed which involve inducing representations from known representations.