Let $A$ be a $n \times n$ matrix and $p$ a complex polynomial. Let $W(A)$ denote the numerical range (or field of values) of a matrix; that is, $W(A) = \{\langle Ax, x\rangle: x \in \mathbb{C}^n, \|x\| = 1\}$. M. Crouzeix conjectured that for every polynomial $p$ we have

\[\|p(A)\| \le 2 \sup_{z \in W(A)} |p(z)|.\] In this talk, we give some background on the conjecture, and explain why we consider it in the model space setting. We then turn to Blaschke products, with a focus on properties that we hope will aid our understanding of the conjecture.