Network measures that reflect the most salient properties of complex large-scale networks are in high demand in the network research community. In this paper we adapt a combinatorial measure of negative curvature (also called hyperbolicity) to parametrized finite networks, and show that a variety of biological and social networks are hyperbolic. This hyperbolicity property has strong implications on the higher-order connectivity and other topological properties of these networks. Specifically, we derive and prove bounds on the distance among shortest or approximately shortest paths in hyperbolic networks. We describe two implications of these bounds to crosstalk in biological networks, and to the existence of central, influential neighborhoods in both biological and social networks.