A Gaussian graphical model is the set of multivariate normal distributions satisfying conditional independence relations specified by an undirected graph, whose vertices are the random variables. In certain applied settings, particularly in genomics, one hopes to fit a Gaussian graphical model to data when there are far fewer data points than random variables. This raises the question: for a given graph G, what is the minimum number of data points required to ensure existence (almost surely) of the maximum likelihood estimate in the corresponding graphical model? This number is called the maximum likelihood threshold of G. In this talk, I will discuss how tools from rigidity theory can be used to understand maximum likelihood thresholds. This is joint work with Sean Dewar, Steven Gortler, Tony Nixon, Meera Sitharam, and Louis Theran.