Abstract: Random topology is the study of topological invariants of random topological spaces. In this talk I will briefly survey work on topology of random simplicial complexes, starting with 1-dimensional models, i.e. random graphs. The Erdos-Renyi theorem characterizes the threshold edge probability where the random graph becomes connected, and we now know several different generalizations of this theorem to higher dimensions. In this talk, I'll discuss recent progress in proving such theorems by understanding eigenvalues of random matrices. I will not assume any particular topology or probability prerequisites, and the talk will aim to be self contained. Part of this is joint work with Chris Hoffman and Elliot Paquette.