  The combination of a finite time-scale for star formation, rapid early
stellar evolution and rapid stellar-dynamical processes imply that the stellar
IMF cannot be inferred for any star cluster independently of its age (the
Cluster IMF Theorem). The IMF can nevertheless be constrained statistically by
evolving many theoretical populations drawn from one parent distribution and
testing these against observed populations. It follows that all known
well-resolved stellar populations are consistent with having been drawn from
the same parent mass distribution. The IMF Universality Hypothesis therefore
cannot be discarded despite the existence of the Cluster IMF Theorem. This
means that the currently existing star-formation theory fails to describe the
stellar outcome, because it predicts a dependency of the IMF on the physical
boundary conditions not observed. The IGIMF Theorem, however, predicts a
variation of galaxy-wide IMFs in dependence of the galaxy's star-formation rate
even if the IMF Universality Hypothesis is valid. This variation has now been
observed in SDSS galaxy data. Detailed analysis of the binary properties in the
very-low-mass star and brown dwarf (BD) mass regime on the one hand, and in the
stellar regime on the other, shows there to be a discontinuity in the IMF near
0.1Msun such that BDs follow a separate distribution function. Very recent
observations of the stellar population within 1pc of the nucleus of the MW do
suggest a top-heavy IMF, perhaps hinting at a variation of the star-formation
outcome with tidal field and temperature thereby violating the IMF Universality
Hypothesis under these physically extreme conditions. Another violation of this
hypothesis appears to emerge for extremely metal-poor stars such that the
primordial IMF appears to have been depleted in low-mass stars.
