(abridged) The {stellar IMF} has been found to be essentially invariant. While some apparent differences are seen, the uncertainties inherent to this game do not allow a firm conclusion to be made that the IMF varies systematically with conditions. The IMF integrated over entire galaxies, however, is another matter. Chemical and photometric properties of various galaxies do hint at {galaxial IMFs} being steeper than the stellar IMF, as is also deduced from direct star-count analysis in the MW. These results are sensitive to the modelling of stellar populations and to corrections for stellar evolution, and are thus also uncertain. However, by realising that galaxies are made from dissolving star clusters, star clusters being viewed as {the fundamental building blocks of galaxies}, the result is found that galaxial IMFs must be significantly steeper than the stellar IMF, because the former results from a folding of the latter with the star-cluster mass function. Furthermore, this notion leads to the important insight that galaxial IMFs must vary with galaxy mass, and that the galaxial IMF is a strongly varying function of the star-formation history for galaxies that have assembled only a small mass in stars. Cosmological implications of this are discussed.

13 pages, to appear in IMFat50: The Initial Mass Function 50 years later, ed: E. Corbelli, F. Palla, and H. Zinnecker, Kluwer Academic Publishers; a meeting held at the Abbazia di Spineto, Tuscany, Italy -- May 16-20, 2004