I discuss "simple" dynamical systems on networks and examine how network structure affects processes that run on top of networks. I'll give an introduction to the idea of a social ("complex") contagion, and I'll present a model for multi-stage complex contagions, in which fanatics produce greater influence than mere followers. I'll also briefly discuss the use of ideas from topics like persistent homology to examine wavefront propagation versus the appearance of new contagion clusters. Time permitting, I'll present a model (without network structure) for the adoption of applications on Facebook. The last family of models illustrates how very different time-dependent dynamics can produce quantitatively similar long-time behavior, which poses both very serious challenges and exciting opportunities for the modeling of complex systems.