Solutions of degenerate nonlinear diffusion equations such as the porous media equation have the property of finite speed of propagation of disturbances. In this talk we review a class of more degenerate equations characterized by vanishing speed of propagation. We explain how an example of such an equation arises as the singular limit of a parabolic equation describing the dynamics of glasslike fluids. This naturally leads to the problem of finding asymptotic formulas for the velocities of diffuse free boundaries. This is work in progress with Roberto Benzi (Dept. Physics, Univ. of Roma Tor Vergata) and Francesco Deangelis (GSSI, L'Aquila).