Though precise measurements of the cosmic microwave background establish inflation as the leading paradigm of the early Universe, the exact microphysics that realize inflation and the subsequent reheating of the Universe are unknown. I will discuss models of the early Universe and the role that observations of stochastic gravitational wave backgrounds could play in constraining them. In particular, I will present the dynamics and signatures of reheating models that lead to the resonant production of Abelian and non-Abelian gauge bosons. I will then show that gravitational wave emission constrains a similar model invoked to resolve the discrepancy in measurements of the present-day expansion rate. Finally, I will discuss the imprint of primordial non-Gaussianity on gravitational waves that are induced by scalar metric fluctuations at second order in perturbation theory.