In this talk, I will discuss a recent development on the use of probabilistic ideas and tools in the study of dispersive PDEs. Following the work of Lebowitz, Rose, and Speer on the construction of Gibbs measures for 1-d NLS, Bourgain proved their invariance under the dynamics of certain Hamiltonian PDES such as NLS, KdV, etc. First, I will go over Bourgain's idea and discuss how it was further developed in recent years. In particular, I will discuss probabilistic Cauchy theory both locally in time and globally in time and immediate implications of theorems in probability to these PDE results.