Abstract: I will review the well-posedness and stability theory of gaseous fluids governed by compressible Euler or Euler-Poisson equations in the presence of physical vacuum free boundaries, and present recent joint work with Jiaqi Liu and Nader Masmoudi on self-similar waiting-time solutions exhibiting a transition from smooth to Hölder regularity and the emergence of moving vacuum boundaries. I will also discuss the construction of a large class of initially smooth, square-integrable solutions that develop gradient blowup near the boundary in finite time through a stability analysis of self-similar solutions.