What does the clustering of a data set tell us about the structure of the distribution from which the data was generated? In this talk I will discuss the sense in which a graph clustering algorithm can be said to recover the "correct" clusters of a random graph distribution. Our modeling assumption is that graphs are sampled from a graphon, which is a powerful and general model for generating graphs and analyzing large networks. Graphons are a far richer class of graph models than stochastic blockmodels, the primary setting for recent progress in the statistical theory of graph clustering. We define what it means for an algorithm to produce the "correct" clustering, give sufficient conditions in which a method is statistically consistent, and provide an explicit algorithm satisfying these properties.