The black hole mass function of supermassive black holes describes the
evolution of the distribution of black hole mass. It is one of the primary
empirical tools available for mapping the growth of supermassive black holes
and for constraining theoretical models of their evolution. In this review we
discuss methods for estimating the black hole mass function, including their
advantages and disadvantages. We also review the results of using these methods
for estimating the mass function of both active and inactive black holes. In
addition, we review current theoretical models for the growth of supermassive
black holes that predict the black hole mass function. We conclude with a
discussion of directions for future research which will lead to improvement in
both empirical and theoretical determinations of the mass function of
supermassive black holes.Comment: 40 pages, 7 figures, review paper accepted for the Advances in
Astronomy Special Issue "Seeking for the Leading Actor on the Cosmic Stage:
Galaxies versus Supermassive Black Holes
