  We critically review the role of cosmological moduli in determining the
post-inflationary history of the universe. Moduli are ubiquitous in string and
M-theory constructions of beyond the Standard Model physics, where they
parametrize the geometry of the compactification manifold. For those with
masses determined by supersymmetry breaking this leads to their eventual decay
slightly before Big Bang Nucleosynthesis (without spoiling its predictions).
This results in a matter dominated phase shortly after inflation ends, which
can influence baryon and dark matter genesis, as well as observations of the
Cosmic Microwave Background and the growth of large-scale structure. Given
progress within fundamental theory, and guidance from dark matter and collider
experiments, non-thermal histories have emerged as a robust and theoretically
well-motivated alternative to a strictly thermal one. We review this approach
to the early universe and discuss both the theoretical challenges and the
observational implications.
