In this paper we show that complex (scale-free) network topologies naturally emerge from hyperbolic metric spaces. The hyperbolic geometry can be used to facilitate maximally efficient greedy forwarding on these topologies where packets can find their destinations with 100% probability following almost optimal, i.e., shortest paths, without the need for global topology knowledge. We demonstrate that this remarkable efficiency is robust under dynamic network conditions. Our findings suggest that forwarding information through complex networks, e.g., like the Internet, may be possible without the current overhead of routing protocols, and may also find practical applications in overlay networks for tasks such as application-level routing, information sharing, and data distribution.