In many dispersive equations, such as the KdV or Benjamin-Ono equations, the dispersive terms appear linearly. There are other equations, however, in which the dispersion enters nonlinearly; examples come from numerical analysis and geophysical applications, among other places. Equations with nonlinear, degenerate dispersion are known to exhibit a variety of non-analytic traveling waves, such as compactons, but for many such equations, the well-posedness theory has not been developed. We demonstrate both ill-posedness and well-posedness results for equations with degenerate dispersion; the well-posedness results are related to a study of the competition between dispersion and backwards diffusion in linear PDE, which we will discuss. This includes joint work with Gideon Simpson, J. Douglas Wright, and Dennis G. Yang.