In this talk we will first give a quick background overview of Bourgain's approach to prove the invariance of the Gibbs measure for the periodic cubic nonlinear Schrodinger equation in 2D and of the para-controlled calculus of Gubinelli-Imkeller and Perkowski in the context of parabolic stochastic equations. We will then present our resolution of the long-standing problem of proving almost sure global well-posedness (i.e. existence with uniqueness) for the periodic nonlinear Schrödinger equation (NLS) in 2D on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is achieved by a new method we call random averaging operators which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is joint work with Yu Deng (USC) and Haitian Yue (USC).