  The modern phase diagram of strongly interacting matter reveals a rich
structure at high-densities due to phase transitions related to the chiral
symmetry of quantum chromodynamics (QCD) and the phenomenon of color
superconductivity. These exotic phases have significant impacts on high-density
astrophysics as the properties of neutron stars and the evolution of
astrophysical systems as proto-neutron stars, core-collapse supernovae and
neutron star mergers. Most recent pulsar mass measurements and constraints on
neutron star radii are critically discussed. Astrophysical signals for exotic
matter and phase transitions in high-density matter proposed recently in the
literature are outlined. A strong first order phase transition leads to the
emergence of a third family of compact stars besides white dwarfs and neutron
stars. The different microphysics of quark matter results in an enhanced r-mode
stability window for rotating compact stars compared to normal neutron stars.
Future telescope and satellite data will allow to extract signals from phase
transitions in dense matter in the heavens and will reveal properties of the
phases of dense QCD. Spectral line profiles out of x-ray bursts will determine
the mass-radius ratio of compact stars. Gravitational wave patterns from
collapsing neutron stars or neutron star mergers will even be able to constrain
the stiffness of the quark matter equation of state. Future astrophysical data
can therefore provide a crucial cross-check to the exploration of the QCD phase
diagram with the heavy-ion program of the CBM detector at the FAIR facility.
