In this paper I present a general framework for regression in the presence of complex dependence structures between units such as in time-series cross-sectional data, relational/network data, and spatial data. These types of data are challenging for standard multilevel models because they involve multiple types of structure (e.g. temporal effects and cross-sectional effects) which are interactive. I show that interactive latent factor models provide a powerful modeling alternative that can address a wide range of data types. Although related models have previously been proposed in several different fields, inference is typically cumbersome and slow. I introduce a class of fast variational inference algorithms that allow for models to be fit quickly and accurately.