This book provides a modern introduction to the representation theory of finite groups. The theory is developed in terms of modules, since this is appropriate for more advanced work, but considerable emphasis is placed upon constructing characters. The character tables of many groups are given, including all groups of order less than 32, and all but one of the simple groups of order less than 1000. Amongst those applications covered are Burnside's paqb theorem, the use of character theory in studying subgroup structure, and a description of how to use representation theory to investigate molecular vibration. Each chapter is accompanied by a variety of exercises, and full solutions to all the exercises are provided at the end of the book. This will be suitable as a text for those teaching a course in representation theory, and in view of the applications of the subject, will be of interest to chemists and physicists as well as mathematicians.