  From time to time, and quite more frequently in recent years, claims appear
favoring a variable Initial Mass Function (IMF), one way or another, either in
time or space. In this chapter we add our two pennies of wisdom, illustrating
how the IMF affects various properties of galaxies and galaxy clusters. We
start by showing that even relatively small variations of the IMF slope have
large effects on the demography of stellar populations, moving the bulk of the
stellar mass from one end to the other of the distribution. We then point out
how the slope of the IMF in different mass ranges controls specific major
properties of galaxies and clusters. The slope of the IMF below ~1 solar mass
controls the M/L ratio of local ellipticals, whereas the slope between ~1 and
~1.4 solar masses controls the evolution with redshift of such ratio, hence of
the fundamental plane of elliptical galaxies. Similarly, the slope between ~1
and ~40 solar masses drives the ratio of the global metal mass in clusters of
galaxies to their total luminosity. While we believe that it is perfectly
legitimate to entertain the notion that the IMF may not be universal, our
message is that when proposing IMF variations to ease a specific problem then
one should not neglect to explore the full consequences of the invoked
variations.
  This paper is integrally reproduced from Chapter 8 of the book by L. Greggio
and A. Renzini: Stellar Populations. A User Guide from Low to High Redshift
(2011, Wiley-VHC Verlag-GmbH & Co., ISBN 9783527409181), whose index is also
appended.
