The paper develops a geometric framework to study the structure and function of complex networks, assuming that hyperbolic geometry underlies these networks. It shows that heterogeneous degree distributions and strong clustering in complex networks naturally emerge as simple reflections of the negative curvature and metric properties of hyperbolic geometry. Conversely, if a network has a metric structure and a heterogeneous degree distribution, it has an effective hyperbolic geometry underneath. The authors establish a mapping between their geometric framework and statistical mechanics of complex networks, interpreting edges as non-interacting fermions with energies equal to hyperbolic distances between nodes. The geometric network ensemble includes the standard configuration model and classical random graphs as limiting cases. They also show that targeted transport processes in networks with strong heterogeneity and clustering achieve the best possible efficiency, which is robust even under catastrophic disturbances. The paper provides a detailed mathematical derivation and analysis of these findings, including the degree distribution, clustering, and phase transitions in the network ensemble.The paper develops a geometric framework to study the structure and function of complex networks, assuming that hyperbolic geometry underlies these networks. It shows that heterogeneous degree distributions and strong clustering in complex networks naturally emerge as simple reflections of the negative curvature and metric properties of hyperbolic geometry. Conversely, if a network has a metric structure and a heterogeneous degree distribution, it has an effective hyperbolic geometry underneath. The authors establish a mapping between their geometric framework and statistical mechanics of complex networks, interpreting edges as non-interacting fermions with energies equal to hyperbolic distances between nodes. The geometric network ensemble includes the standard configuration model and classical random graphs as limiting cases. They also show that targeted transport processes in networks with strong heterogeneity and clustering achieve the best possible efficiency, which is robust even under catastrophic disturbances. The paper provides a detailed mathematical derivation and analysis of these findings, including the degree distribution, clustering, and phase transitions in the network ensemble.