The Dawes Review 8: Measuring the Stellar Initial Mass Function
  The birth of stars and the formation of galaxies are cornerstones of modern
astrophysics. While much is known about how galaxies globally and their stars
individually form and evolve, one fundamental property that affects both
remains elusive. This is problematic because this key property, the birth mass
distribution of stars, referred to as the stellar initial mass function (IMF),
is a key tracer of the physics of star formation that underpins almost all of
the unknowns in galaxy and stellar evolution. It is perhaps the greatest source
of systematic uncertainty in star and galaxy evolution. The past decade has
seen a growing number and variety of methods for measuring or inferring the
shape of the IMF, along with progressively more detailed simulations,
paralleled by refinements in the way the concept of the IMF is applied or
conceptualised on different physical scales. This range of approaches and
evolving definitions of the quantity being measured has in turn led to
conflicting conclusions regarding whether or not the IMF is universal. Here I
review and compare the growing wealth of approaches to our understanding of
this fundamental property that defines so much of astrophysics. I summarise the
observational measurements from stellar analyses, extragalactic studies and
cosmic constraints, and highlight the importance of considering potential IMF
variations, reinforcing the need for measurements to quantify their scope and
uncertainties carefully, in order for this field to progress. I present a new
framework to aid the discussion of the IMF and promote clarity in the further
development of this fundamental field.
