The construction of magnetohydrodynamic equilibria in non-axisymmetric toroidal domains is a notoriously challenging problem. The existence of solutions for smooth pressure profiles has not been proved, and empirical results suggest that non-integrable singularities for the current density are generically obtained for such pressure profiles. In this presentation, I will describe the mathematical nature of the difficulties, discuss the special cases in which proofs of existence do exist, and present preliminary theoretical and numerical results which suggest a promising direction for a proof of existence and the numerical construction of equilibria in a robust manner.