  Recent inference results of the sound velocity in the cores of neutron stars
are summarized. Implications for the equation of state and the phase structure
of highly compressed baryonic matter are discussed. In view of the strong
constraints imposed by the heaviest known pulsars, the equation of state must
be very stiff in order to ensure the stability of these extreme objects. This
required stiffness limits the possible appearance of phase transitions in
neutron star cores. For example, a Bayes factor analysis quantifies strong
evidence for squared sound velocities $c_s^2 > 0.1$ in the cores of 2.1
solar-mass and lighter neutron stars. Only weak first-order phase transitions
with a small phase coexistence density range $\Delta\rho/\rho < 0.2$ (at the
68\% level) in a Maxwell construction still turn out to be possible within
neutron stars. The central baryon densities in even the heaviest neutron stars
do not exceed five times the density of normal nuclear matter. In view of these
data-based constraints, much discussed issues such as the quest for a phase
transition towards restored chiral symmetry, and the active degrees of freedom
in cold and dense baryonic matter, are reexamined.
