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 M _ { \odot } 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 \approx 2 .
These two results imply that the integrated galactic IMF ( IGIMF ) for early-type stars can not 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 .