  The possible role of a first order QCD phase transition at nonvanishing quark
chemical potential and temperature for cold neutron stars and for supernovae is
delineated. For cold neutron stars, we use the NJL model with nonvanishing
color superconducting pairing gaps, which describes the phase transition to the
2SC and the CFL quark matter phases at high baryon densities. We demonstrate
that these two phase transitions can both be present in the core of neutron
stars and that they lead to the appearance of a third family of solution for
compact stars. In particular, a core of CFL quark matter can be present in
stable compact star configurations when slightly adjusting the vacuum pressure
to the onset of the chiral phase transition from the hadronic model to the NJL
model. We show that a strong first order phase transition can have strong
impact on the dynamics of core collapse supernovae. If the QCD phase transition
sets in shortly after the first bounce, a second outgoing shock wave can be
generated which leads to an explosion. The presence of the QCD phase transition
can be read off from the neutrino and antineutrino signal of the supernova.
