  Over the past years observations of young and populous star clusters have
shown that the stellar initial mass function (IMF) can be conveniently
described by a two-part power-law with an exponent alpha_2 = 2.3 for stars more
massive than about 0.5 Msol and an exponent of alpha_1 = 1.3 for less massive
stars. A consensus has also emerged that most, if not all, stars form in
stellar groups and star clusters, and that the mass function of these can be
described as a power-law (the embedded cluster mass function, ECMF) with an
exponent beta ~2. These two results imply that the integrated galactic IMF
(IGIMF) for early-type stars cannot be a Salpeter power-law, but that they must
have a steeper exponent. An application to star-burst galaxies shows that the
IGIMF can become top-heavy. This has important consequences for the
distribution of stellar remnants and for the chemo-dynamical and photometric
evolution of galaxies. In this contribution the IGIMF theory is described, and
the accompanying contribution by Pflamm-Altenburg, Weidner & Kroupa (this
volume) documents the applications of the IGIMF theory to galactic
astrophysics.
