  We discuss the physics of moduli (light scalar fields with Planck-suppressed
couplings to matter) in the case of low-scale supersymmetry breaking such as
gauge mediation. We argue that even if the mechanism of moduli stabilization is
decoupled from the mechanism of SUSY breaking, moduli masses will generically
be parametrically related to the gravitino mass once the cancellation of the
cosmological constant is taken into account. For low- scale SUSY breaking, this
implies that moduli fields are light, long-lived relics that will generically
drive the universe into a matter-dominated phase, in contradiction to standard
BBN. We discuss two scenarios for evading this problem. The first is to
consider very tuned supergravity potentials that can make the moduli heavy
enough to decay at early times and reheat above the temperature of BBN. Viable
cosmology can be achieved in this scenario, which has a population of highly
relativistic light gravitinos arising from decays of moduli. Next, we consider
the more natural scenario with light moduli. The saxion field associated with
the solution of the strong CP problem provides the most natural candidate for
driving thermal inflation, which dilutes the moduli. We construct an explicit
model for this scenario, when the PQ symmetry breaking scale is larger than the
messenger scale. The combination of the constraints on relic abundance and
gamma-ray flux from decay of the moduli favors a particular region of SUSY
breaking scale, 2 x 10^3 - 10^4 TeV. For either scenario, we find that it is
generic for low-scale SUSY to be associated with late entropy production and
accompanying low reheating temperatures.
