In this talk we will use characters on the unitary group to describe properties of the trace and eigenvalues of random unitary matrices. We will use the group structure of U(n) to take questions about random matrices and rephrase them as questions on the inner product of class functions and thus characters. Little to no knowledge of probability will be assumed. This talk is based on the work of Persi Diaconis ad Mehrdad Shahshahani.