Symmetric edge polytopes, also known as adjacency polytopes, are centrally symmetric lattice polytopes that encode the edge set of a finite simple graph. These polytopes have been the subject of extensive recent study due to connections with both pure and applied mathematics. I will provide a survey of results regarding triangulations, Ehrhart polynomials, and halfspace-descriptions for these polytopes. I will then discuss recent joint work with Kaitlin Bruegge and Matthew Kahle regarding facet numbers of symmetric edge polytopes for graphs with few vertices and for various families of random graphs.