  We use state-of-the-art chemical models to track the cosmic evolution of the
CNO isotopes in the interstellar medium (ISM) of galaxies, yielding powerful
constraints on their stellar initial mass function (IMF). We re-assess the
relative roles of massive stars, asymptotic giant branch (AGB) stars and novae
in the production of rare isotopes such as 13C, 15N, 17O and 18O, along with
12C, 14N and 16O. The CNO isotope yields of super-AGB stars, novae and
fast-rotating massive stars are included. Having reproduced the available
isotope enrichment data in the solar neighbourhood, and across the Galaxy, and
having assessed the sensitivity of our models to the remaining uncertainties,
e.g. nova yields and star-formation history, we show that we can meaningfully
constrain the stellar IMF in galaxies using C, O and N isotope abundance
ratios. In starburst galaxies, where data for multiple isotopologue lines are
available, we find compelling new evidence for a top-heavy stellar IMF, with
profound implications for their star-formation rates and efficiencies, perhaps
also their stellar masses. Neither chemical fractionation nor selective
photodissociation can significantly perturb globally-averaged isotopologue
abundance ratios away from the corresponding isotope ones, as both these
processes will typically affect only small mass fractions of molecular clouds
in galaxies. Thus the Atacama Large Millimetre Array now stands ready to probe
the stellar IMF, and even the ages of specific starburst events in star-forming
galaxies across cosmic time unaffected by the dust obscuration effects that
plague optical/near-infrared studies.
