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.

Invited contribution to the EPJA Topical Issue "Exotic Matter in Neutron Stars", 11 pages, 6 figures