  We review some recent results about the mechanism of deconfinement of
hadronic matter into quark matter in cold neutron stars and protoneutron stars.
We discuss the role of finite size effects and the relevance of temperature and
density fluctuations on the nucleation process. We also examine the importance
of surface effects for mixed phases in hybrid stars. A small drop of quark
matter nucleated at the core of a compact star may grow if the conversion is
sufficiently exothermic. In such a case, it may trigger the burning of the
stellar core and even the whole star if quark matter is absolutely stable. We
explore the physical processes that occur inside the flame and analyze the
hydrodynamic evolution of the combustion front. In the last part of this
review, we focus on hybrid stars using the Nambu-Jona-Lasinio (NJL) model with
scalar, vector and 't Hooft interactions, paying particular attention to a
generalized non-standard procedure for the choice of the 'bag constant'. We
also describe the non-radial oscillation modes of hadronic, hybrid and strange
stars with maximum masses above $2 M_{\odot}$ and show that the frequency of
the $p_1$ and $g$ fluid modes contains key information about the internal
composition of compact objects.
