Methods for descriptive network analysis have reached statistical maturity and general acceptance across the social sciences in recent years. However, methods for statistical inference with network data remain fledgling by comparison. We introduce and evaluate a general model for inference with network data, the Exponential Random Graph Model (ERGM) and several of its recent extensions. The ERGM simultaneously allows both inference on covariates and for arbitrarily complex network structures to be modeled. Our contributions are three-fold: beyond introducing the ERGM and discussing its limitations, we discuss extensions to the model that allow for the analysis of non-binary and longitudinally observed networks and show through applications that network-based inference can improve our understanding of political phenomena.