I will describe some cohomology theories of topological automorphic forms which arise from a theorem of Jacob Lurie. These are associated to Shimura varieties for the groups U(1, n − 1) in the same way that topological modular forms are associated to the moduli space of elliptic curves. The corresponding theories detect vn-periodic phenomena. I will also describe a related spectrum which approximates the K(n)-local sphere. This is joint work with Tyler Lawson.