The nonequilibrium dynamics of quantum fields is studied in inflationary cosmology, with particular emphasis on applications to the problem of post-inflation reheating. The Schwinger-Keldysh closed-time-path (CTP) formalism is utilized along with the two-particle-irreducible (2PI) effective action in order to obtain coupled, nonperturbative equations for the mean field and variance in a general curved background spacetime. For a model consisting of a quartically self-interacting O(N) field theory (with unbroken symmetry) in spatially flat FRW spacetime, the dynamics of the mean field is studied numerically, at leading order in the large-N expansion. The time evolution of the scale factor is determined self-consistently using the semiclassical Einstein equation. It is found that cosmic expansion can dramatically affect the efficiency of parametric resonance-induced particle production. The production of fermions due to the oscillating inflaton mean field is studied for the case of a scalar inflaton coupled to a fermion field via a Yukawa coupling $f$. The dissipation and noise kernels appearing at $O(f^2)$ in the one-loop CTP effective action are shown to satisfy a zero-temperature fluctuation-dissipation relation (FDR). The effective stochastic equation obeyed by the inflaton zero mode at $O(f^4)$ contains multiplicative noise. It is shown that stochasticity becomes important to the dynamics of the inflaton zero mode before the end of reheating. The thermalization problem is discussed, and a strategy is presented for obtaining time-local equations for equal-time correlation functions which goes beyond the Hartree-Fock approximation.