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Abstract: The sigma_k Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, in which one looks to prescribe symmetric functions of the eigenvalues of the Schouten tensor to be constant within a fixed conformal class. In the last 20 years or so, there has been a significant amount of progress on the sigma_k Yamabe problem in the so-called positive case, leading to many existence results for smooth solutions. In this talk, I will discuss some recent results on the regularity theory for the sigma_k Yamabe equation, with an emphasis on W^{2,p} solutions in both the positive and negative cases, and viscosity solutions to the degenerate problem. I will also mention some open problems in the field. Much of the talk will be based on joint work with Luc Nguyen.