Erdos and Renyi introduced a model for studying random graphs of a given density and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will describe new threshold theorems for random graphs and applications of these results to the geometry of Coxeter groups. This talk will include joint work with Falgas-Ravry, Hagen, Sisto, and Susse (in various combinations).