a pervading question in the study of stochastic pde is how small-scale random forcing in an equation combines to create nontrivial statistical behavior on large spatial and temporal scales. i will discuss recent progress on this topic for several related stochastic pdes - stochastic heat, kpz, and burgers equations - and some of their generalizations. these equations are (conjecturally) universal models of physical processes such as a polymer in a random environment, the growth of a random interface, branching brownian motion, and the voter model. the large-scale behavior of solutions on large scales is complex, and in particular depends qualitatively on the dimension of the space. i will describe the phenomenology, and then describe several results and challenging problems on invariant measures, growth exponents, and limiting distributions.