This book presents the notes for a course given on the theory of unitary group representations at the University of Chicago in 1955. This new revision represents the small but steady demand for these notes, which have been reviewed, proofs have been clarified, and the term "locally simple" has been replaced by the more appropriate term "multiplicity free." This twenty-year old material has been augmented by an appendix which sketches the developments which have taken place between 1955 and 1975, as well as the early history of the subject.