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.

40 pages, 20 figures. To appear in Springer volume on "Harmonic Analysis and Applications", February 2021. Corrected typos and made a notational change: the regularization parameter is now called $\varepsilon$ instead of $x$