One of the most effective tools for proving that Strong Cosmic Censorship holds for a family of cosmological solutions of Einstein’s equations is to verify that these solutions all exhibit Asymptotically Velocity Term Dominated (AVTD) behavior in a neighborhood of their Big Bang singularities. After presenting some background history of known results and conjectures concerning the behavior of the gravitational field near the Big Bang in cosmological solutions of Einstein’s equations, we discuss recent work with Ellery Ames, Florian Beyer, and Todd Oliynyk in which we prove that polarized $T^2$-Symmetric vacuum solutions in a neighborhood of Kasner solutions all exhibit AVTD behavior close to the initial singularity. We also discuss our very recent extension of these results to the case of non-vanishing cosmological constant.