In this talk, we discuss recent work on Gromov's delta-hyperbolicity in the context of random graph models, tree-decompositions, and empirical evaluation of network structure. Specifically, we characterize when random intersection graphs have bounded hyperbolicity, give general theoretical bounds on delta in terms of tree-width, and describe a relationship to local measurement of the core structure. More generally, we describe empirical results on tree-like structure in complex networks and suggest several open problems. This is joint work with various subsets of Aaron Adcock, Matthew Farrell, Timothy Goodrich, Nathan Lemons, Michael Mahoney, Michael O'Brien, and Felix Reidl.