We discuss the role of differentiability in the regularity of the solutions to homogeneous, fully nonlinear elliptic equations. The first part of the chapter concerns flat solutions to equations driven by operators of class $C^2$. Owing to Savin, this result ensures that solutions are of class $C^{2,\alpha}$ and depend only on uniform ellipticity in a neighborhood of the origin in the space of symmetric matrices. We highlight the relevance of differentiability as a condition replacing convexity in the Evans–Krylov theory. The second part of the chapter details the partial regularity result, which prescribes an upper bound for the Hausdorff dimension of the singular set for the viscosity solutions to $F(D^2u)=0$. This result cleverly combines the smoothness of flat solutions and Lin's integral estimates.