Since the 1920's, it has been known that the spatially homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes generically develop curvature singularities in the contracting time direction along spacelike hypersurfaces, known as big bang singularities, both in vacuum and for a wide range of matter models. For many years, it remained unclear if the big bang singularities were physically relevant. It was thought by some that big bang singularities were due to the unphysical assumption of spatial homogeneity and that they would disappear in non-homogenous spacetimes, or in other words, big bang singularities were unstable under nonlinear perturbations as solutions to the Einstein field equations. A partial resolution to this situation came in 1967 when Hawking established his singularity theorem that guarantees a cosmological spacetime will be geodesically incomplete for a large class of matter models and initial data sets, including highly anisotropic ones. While Hawking's singularity theorem guarantees that cosmological spacetimes are geodesically incomplete (i.e. at least one observer will experience something pathological at a finite time in the past) for a large class of initial data sets, it is silent on the cause of the geodesic incompleteness. It has been widely anticipated that the geodesics incompleteness is due to the formation of curvature singularities, and it is an outstanding problem in mathematical cosmology to rigorously establish the conditions under which this expectation is true and to understand the dynamical behaviour of cosmological solutions near singularities.

In this talk, I will begin by introducing the FLRW and Kasner solutions of the Einstein-scalar field equations, which are exact, spatially homogeneous solutions that play a distinguished role in the analysis of big bang singularities. After briefly providing context for the FLRW and Kasner solutions in the historical development of the field of cosmology,
I will define what it means for a FLRW/Kasner big bang singularity to be stable. With this notion in hand, I will then discuss the recent influential FLRW and Kasner big bang stability proofs of Rodnianski-Speck and Fournodavlos-Rodnianski-Speck. One aspect of these stability results that I will pay particular attention to is their global nature. To conclude the talk, I will discuss some recent work done in collaboration with Florian Beyer where we improve the Rodnianski-Speck FLRW big bang stability result by establishing that the FRLW big bang is locally stable, which is a significantly stronger notion of stability with important physical consequences that I will briefly discuss. Time permitting, I will also briefly discuss more recent work with Florian Beyer on the stability of big bang singularities when relativistic fluids are present.