  We survey the current status of universality limits for $m$-point correlation
functions in the bulk and at the edge for unitary ensembles, primarily when the
limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider
underlying measures on compact intervals, and fixed and varying exponential
weights, as well as universality limits for a variety of orthogonal systems.
The scope of the survey is quite narrow: we do not consider $\beta$ ensembles
for $\beta \neq 2$, nor general Hermitian matrices with independent entries,
let alone more general settings. We include some open problems.
