The influence of confinement on the freezing transition of hard spheres is investigated. Two limiting cases are considered: (1) large systems, where walls weakly perturb the bulk system, and (2) small systems where the influence of geometry becomes important. In the first situation, the shift in coexisting densities is a linear function of the area to volume ratio in the system: This is a manifestation of the Kelvin equation, and the phenomenon is thermodynamically equivalent to capillary condensation. A claim (by others) of "prefreezing" of hard spheres at a smooth hard wall is quantitatively attributed to capillary crystallization. It is shown that the coexistence region narrows as a function of the area to volume ratio. In the second limit two different confined geometries are studied. In these limits, widening of the coexistence region is observed, pointing to an upper and lower critical point at intermediate values of the area to volume ratio, or no critical point at all. In a slit geometry buckling transitions interfere with the freezing transition. In a box geometry, at large values of the area to volume ratio, fluctuations become important. These fluctuations determine the fate of the freezing transition at intermediate values of the area to volume ratio.