  We derive the theoretical distribution function of black hole masses by
studying the formation processes of black holes. We use the results of recent
2D simulations of core-collapse to obtain the relation between remnant and
progenitor masses and fold it with an initial mass function for the
progenitors. We examine how the calculated black-hole mass distributions are
modified by (i) strong wind mass loss at different evolutionary stages of the
progenitors, and (ii) the presence of close binary companions to the black-hole
progenitors. Thus, we are able to derive the binary black hole mass
distribution. The compact remnant distribution is dominated by neutron stars in
the mass range 1.2-1.6Msun and falls off exponentially at higher remnant
masses. Our results are most sensitive to mass loss from winds which is even
more important in close binaries. Wind mass-loss causes the black hole
distribution to become flatter and limits the maximum possible black-hole mass
(<10-15Msun). We also study the effects of the uncertainties in the explosion
and unbinding energies for different progenitors. The distributions are
continuous and extend over a broad range. We find no evidence for a gap at low
values (3-5Msun) or for a peak at higher values (~7Msun) of black hole masses,
but we argue that our black hole mass distribution for binaries is consistent
with the current sample of measured black-hole masses in X-ray transients. We
discuss possible biases against the detection or formation of X-ray transients
with low-mass black holes. We also comment on the possibility of black-hole
kicks and their effect on binaries.
