  Observational cosmology in the next decade will rely on probes of the
distribution of matter in the redshift range between $0<z<3$ to elucidate the
nature of dark matter and dark energy. In this redshift range, galaxy formation
is known to have a significant impact on observables such as two-point
correlations of galaxy shapes and positions, altering their amplitude and scale
dependence beyond the expected statistical uncertainty of upcoming experiments
at separations under 10 Mpc. Successful extraction of information in such a
regime thus requires, at the very least, unbiased models for the impact of
galaxy formation on the matter distribution, and can benefit from complementary
observational priors. This work reviews the current state of the art in the
modelling of baryons for cosmology, from numerical methods to approximate
analytical prescriptions, and makes recommendations for studies in the next
decade, including a discussion of potential probe combinations that can help
constrain the role of baryons in cosmological studies. We focus, in particular,
on the modelling of the matter power spectrum, $P(k,z)$, as a function of scale
and redshift, and of the observables derived from this quantity. This work is
the result of a workshop held at the University of Oxford in November of 2018.
