  White dwarfs are the final remnants of low- and intermediate-mass stars.
Their evolution is essentially a cooling process that lasts for $\sim 10$ Gyr.
Their observed properties provide information about the history of the Galaxy,
its dark matter content and a host of other interesting astrophysical problems.
Examples of these include an independent determination of the past history of
the local star formation rate, identification of the objects responsible for
the reported microlensing events, constraints on the rate of change of the
gravitational constant, and upper limits to the mass of weakly interacting
massive particles. To carry on these tasks the essential observational tools
are the luminosity and mass functions of white dwarfs, whereas the theoretical
tools are the evolutionary sequences of white dwarf progenitors, and the
corresponding white dwarf cooling sequences. In particular, the observed white
dwarf luminosity function is the key manifestation of the white dwarf cooling
theory, although other relevant ingredients are needed to compare theory and
observations. In this review we summarize the recent attempts to empirically
determine the white dwarf luminosity function for the different Galactic
populations. We also discuss the biases that may affect its interpretation.
Finally, we elaborate on the theoretical ingredients needed to model the white
dwarf luminosity function, paying special attention to the remaining
uncertainties, and we comment on some applications of the white dwarf cooling
theory. Astrophysical problems for which white dwarf stars may provide useful
leverage in the near future are also discussed.
