Integrable equations have large classes of solutions that are parametrized by compact Riemann surfaces. These are the so-called finite-genus solutions. The stability of these solutions has not been examined much, until recently. Extending and generalizing earlier work by McKean, we have been able to prove the orbital stability of the finite-genus solutions of the KdV equation with respect to so-called subharmonic perturbations: perturbations whose period is an integer multiple of the period of the finite-genus solution. Our methods extend easily to other integrable equations such as the modified KdV equation and the NLS equation. I will outline the general method, using specific examples along the way. I will conclude with some conjectures.