The authors develop a geometric framework to study the structure and function of complex networks. They assume that hyperbolic geometry underlies these networks, and they show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, they show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. They then establish a mapping between their geometric framework and statistical mechanics of complex networks.