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I review recent progress in understanding the dynamics of light self-interacting scalar fields in de Sitter space at low (superhorizon) momenta. In this regime, perturbation theory is plagued by secular/infrared divergences due to the gravitational amplification of long wavelength quantum fluctuation and a correct description requires resummation schemes or nonperturbative tools. I present recent results based on large-N techniques, solutions of the Dyson-Schwinger equations, and nonperturbative renormalization group techniques.