The goals for the series of talks are as follows. 1. To fill in background for and to state the result of Haagerup, Schultz and Thorbjornsen for polynomials in GUE matrices. 2. To prove the result taking care to explain the most important ``tricks'', e.g, the linearization trick. 3. To discuss extensions and related results in the literature. 4. To prove some extensions of HST to polynomials in Wigner matrices (not the best or sharpest possible) using methods worth learning about in their own right. 5. To point out some open problems in the area.