We consider a broad class of graphical models defined in terms of mixed graphs with directed, undirected, and bidirected edges that can capture sets of independence that arise in various statistical contexts including those in which there is some combination of feedback, latent and selection processes. We introduce the subclass of separable models, those models in which a missing edge corresponds to an independence, and weakly separable models, those models equivalent to some separable graphical model. We characterize separable and weakly separable graphs under two graph separations criterion --- generalized d-separation and $\sigma$-separation --- and show that every equivalence class of separable graphs contains a separable graph with at most one edge between any pair of vertices. We provide multiple characterizations separation equivalence for separable graphs and provide algorithms for testing the equivalence of two separable graphical models and for identifying equivalence classes of separable graphical models.
