  This article begins with a brief review of random matrix theory, followed by
a discussion of how the large-$N$ limit of random matrix models can be realized
using operator algebras. I then explain the notion of "Brown measure," which
play the role of the eigenvalue distribution for operators in an operator
algebra.
  I then show how methods of partial differential equations can be used to
compute Brown measures. I consider in detail the case of the circular law and
then discuss more briefly the case of the free multiplicative Brownian motion,
which was worked out recently by the author with Driver and Kemp.
