  In this survey article we revisit Hilbert's $19^{\text{th}}$ problem
concerning the regularity of minimizers of variational integrals. We first
discuss the classical theory (that is, the statement and resolution of
Hilbert's problem in all dimensions). We then discuss recent results concerning
the regularity of minimizers of degenerate convex functionals. Finally, we
discuss some open problems. Exercises are included for the benefit of
researchers who are entering the subject.
