The stellar initial mass function (IMF) in star clusters is
reviewed. Uncertainties in the observations are emphasized. We
suggest there is a distinct possibility that cluster IMFs vary
systematically with density or pressure. Dense clusters could have
additional formation processes for massive stars that are not
present in low density regions, making the slope of the upper mass
IMF somewhat shallower in clusters. Observations of shallow IMFs
in some super star clusters and in elliptical galaxies are
reviewed. We also review mass segregation and the likelihood that
peculiar IMFs, as in the Arches cluster, result from segregation
and stripping, rather than an intrinsically different IMF. The
theory of the IMF is reviewed in some detail. Several problems
introduced by the lack of a magnetic field in SPH simulations are
discussed. The universality of the IMF in simulations suggests
that something more fundamental than the physical details of a
particular model is at work. Hierarchical fragmentation by any of
a variety of processes may be the dominant cause of the power law
slope. Physical differences from region to region may make a
slight difference in the slope and also appear in the low-mass
turnover point.