Abstract: I will discuss large $N$ asymptotics for $N$-fold integrals of the form \[ Z_{N}=\int_{\mathbb R^N}\prod_{i\neq j}|\lambda_i-\lambda_j|\prod_{j=1}^N e^{-N V(\lambda_j)}d\lambda_j, \] where $V$ is a polynomial. Such integrals appear as partition functions in unitary random matrix models, and can be expressed as Hankel determinants or in terms of orthogonal polynomials on the real line. The nature of the large $N$ asymptotics for $Z_N$ depends on $V$ through an equilibrium problem. Many results about these asymptotics are available in the physics literature, but mathematically rigorous results are only known for a restricted class of polynomials $V$. I will give an overview of known results and indicate some open problems.