  We review the current status of the research on effective nonlocal NJL-like
chiral quark models with separable interactions, focusing on the application of
this approach to the description of the properties of hadronic and quark matter
under extreme conditions. The analysis includes the predictions for various
hadron properties in vacuum, as well as the study of the features of
deconfinement and chiral restoration phase transitions for systems at finite
temperature and/or density. We also address other related subjects, such as the
study of phase transitions for imaginary chemical potentials, the possible
existence of inhomogeneous phase regions, the presence of color
superconductivity, the effects produced by strong external magnetic fields, and
the application to the description of compact stellar objects.
