I will discuss joint work with B. Driver and T. Kemp on the heat equation on the unitary group U(N), in the large-N limit. With an appropriate scaling of the metric, the action of the Laplacian on U(N), at least on certain natural types of functions, has a limit as N tends to infinity. A key point in understanding this limit is to recognize that the Laplacian behaves for large-N like a first-order operator. I will discuss implications of this analysis for the large-N limit of the Segal-Bargmann transform.