The initial mass function of stars and the star-formation rates of
  galaxies
  The measured star-formation rates (SFRs) of galaxies comprise an important
constraint on galaxy evolution and also on their cosmological boundary
conditions. Any available tracer of the SFR depends on the shape of the
mass-distribution of formed stars, i.e. on the stellar initial mass function
(IMF). The luminous massive stars dominate the observed photon flux while the
dim low-mass stars dominate the mass in the freshly formed population. Errors
in the number ratio of the massive to low-mass stars lead to errors in SFR
measurements and thus to errors concerning the gas-accretion-rates and the
gas-consumption time-scales of galaxies. The stellar IMF has traditionally been
interpreted to be a scale-invariant probability density distribution function
(PDF), but it may instead be an optimal distribution function. In the PDF
interpretation, the stellar IMF observed on the stales of individual star
clusters is equal to the galaxy-wide IMF (gwIMF) which, by implication, would
be invariant. In this Chapter we discuss the fundamental properties of the IMF
and of the gwIMF, the nature of both and their systematic variability as
indicated by measurements and theoretical expectations, and we discuss the
implications for the SFRs of galaxies and their main sequence. The importance
of the putative most-massive-star-mass vs
stellar-mass-of-the-hosting-embedded-cluster (mmax-Mecl) relation and its
possible establishment during the proto-stellar formation phase, is emphasised.
