The regularity problem for the three-dimensional Navier-Stokes equations remains unresolved, and is the topic of one of the Clay Foundation's million dollar Millennium Problems. Although singular behaviour has not yet been ruled out, there are some powerful results that restrict the possible size of the singular set, should one actually exist. I will give a fairly detailed but I hope reasonably gentle proof of these results, following for the most part the treatment of Caffarelli, Kohn, & Nirenberg, but with some help from later work of Lin, Ladyzhenskaya & Seregin, and Kukavica. I will then discuss some related results and applications, in particular the construction of Scheffer showing that these partial regularity results are in some sense optimal (given that they apply to any functions satisfying the local energy inequality) and the use of partial regularity to show the almost everywhere uniqueness of Lagrangian trajectories, which was joint work with Witold Sadowski.