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Abstract: A classical problem in PDEs concerns the regularity of stable solutions to nonlinear elliptic equations. This turns out to be a delicate question, even for apparently simple semilinear problems. In this setting, the smoothness of stable solutions depends on the dimension. Namely, for the Laplacian, Cabré, Figalli, Ros-Oton, and Serra (2020) have shown that stable solutions are smooth up to dimension 9, while singularities may appear in higher dimensions. In this talk, I will discuss analogous results for more general operators while giving an overview of the regularity theory for stable solutions.