  Observations have not yielded convincing results concerning the form of the
stellar initial mass function (IMF) or its variations in space and time, so it
is proposed that theoretical models may provide useful guidance. Several
classes of theoretical models for the physics controlling the IMF are reviewed,
emphasizing the dependence of the results on the probability distribution of
velocity and density fluctuations, the spatial density distribution, the
softness of the equation of state as controlled by the dominant heating and
cooling processes, the ambient radiation field, and the star formation rate
itself. One particularly relevant result is the ability of turbulent
interactions to form high-density condensations, which should vary with
decreasing redshift as the dominant cooling mechanism changes from H$_2$, to
fine structure lines, and finally to CO and gas-grain collisions, due to
corresponding changes in the effective polytropic exponent. A second theme is
the ability of some of these models to produce a characteristic or turnover
mass in the IMF which has nothing to do with the Jeans mass. Further
development of such models should provide IMFs whose functional dependence can
be used as input for simulations of galaxy formation, in which the energy input
from massive stars plays a crucial role, and for calculations of the
consequences of protogalactic star formation on the subequent evolution of the
universe. The probable irrelevance of the concepts of pressure-confined clouds
and the formation of condensations by thermal instability in a turbulent
interstellar medium is also discussed.
