For more realistic modeling, there have been significant interests for the use of general graphs for biological models of interacting particle systems, but complex spatial structure often poses severe difficulties to mathematical methods. The fundamental technical issues include the missing link to the stochastic PDE method that can give very detailed and clean information of interacting particle systems after rescaling, and the question of which graph parameters are essential to characterize the probability laws that are of particular interest. In this talk, I will discuss voter models and related methods, with an emphasis on the context of general graphs. I will demonstrate recent progress for a question from Aldous for voter models and mathematical results for evolutionary games that are variants of voter models. In particular, I will explain some diffusion approximation results for voter models on general graphs, which may provide new insights for related interacting particle systems.