The analysis of threshold contagion processes in large networks is challenging. While the lack of accurate network data is often a major obstacle, finding optimal interventions is computationally intractable even in well-measured large networks. To obviate these issues we consider threshold contagion over networks sampled from a graphon—a flexible stochastic network formation model—and show that in this case the contagion outcome can be predicted by only exploiting information about the graphon. To this end, we exploit a second interpretation of graphons as graph limits to formally define a threshold contagion process on a graphon for infinite populations. We then show that contagion in large but finite sampled networks is well approximated by graphon contagion. This convergence result suggests that one can design interventions for large sampled networks by first solving the equivalent problem for an infinite population interacting according to the limiting graphon. We show that, under suitable regularity assumptions, the latter is a tractable problem and we provide analytical characterizations for the extent of contagion and for optimal seeding policies in graphons with both finite and infinite agent types.