I will begin by reviewing de Sitter space, and the problems with perturbative quantization on de Sitter and Starobinsky's stochastic inflation. Next, I will present a non-perturbative technique based on a Legendre transform and show how to construct an effective action from which one can calculate the quantum state of fields at the asymptotic future infinity and thus understand deep infrared (IR) correlators. I will also present a proof that an O(N) symmetric scalar theory, that is broken in the UV, undergoes a symmetry restoration in the deep IR. Namely, the strong quantum fluctuations on de Sitter generate a large amount of vacuum energy, thus restoring the symmetry.