Yang-Mills theory is a key part of the standard model of particle physics. According to the (Euclidean) path-integral formulation of the theory, one is supposed to consider an integral over the space of connections on a principal G-bundle over the relevant manifold, where G is a compact Lie group. If the manifold is the plane, this integral can be understood rigorously as coming from a probability measure on the space of connections, thanks to work of L. Gross, C. King, and A. Sengupta. A more recent development in the physics literature concerns the “large-N limit,” which means one should take G to the unitary group U(N) and then let N tend to infinity. I will discuss recent developments in this area, including work with B. Driver and T. Kemp of UCSD.