I will first discuss existence of families of radial self-similar imploding solutions to the compressible Euler-Poisson and the Einstein-Euler system, representing collapsing stars.

I will then explain the proof of nonlinear radial stability of the (Newtonian) Larson-

Penston solution, which represents a ground state in the above family of collapsing profiles. At the heart of the proof is a difficult non self-adjoint spectral problem and a high-order energy framework which crucially takes advantage of the various monotonicity properties of the Larson-Penston self-similar profile. This is joint work with Y. Guo, J. Jang, and M. Schrecker.