In this talk, we present some of our recent work on the analysis of Boltzmann equations with tools of nonlinear dispersive PDEs. The starting point of our approach is to map the Boltzmann equation, by use of the Wigner transform, to an equation similar to a Schrodinger equation in density matrix formulation with a nonlinear self-interaction. We prove local well-posedness, propagation of moments, and small data global well-posedness in spaces defined by weighted space-time norms of Sobolev type. This is joint work with Ryan Denlinger and Natasa Pavlovic.