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.

to be published in "Massive Stars: From Pop III and GRBs to the Milky Way," eds. M. Livio and Eva Villaver, Cambridge Univ. Press, in press, from a conference held at the Space Telescope Science Institute, May 8-11, 2006. 12 pages