Recent years have shown a promising progress in understanding geometric underpinnings behind the structure, function, and dynamics of many complex networks in nature and society. However, these promises cannot be readily fulfilled and lead to important practical applications, without a simple, reliable, and fast network mapping method to infer the latent geometric coordinates of nodes in a real network. Here, we present HyperMap, a simple method to map a given real network to its hyperbolic space. The method utilizes a recent geometric theory of complex networks modeled as random geometric graphs in hyperbolic spaces. The method replays the network's geometric growth, estimating at each time-step the hyperbolic coordinates of new nodes in a growing network by maximizing the likelihood of the network snapshot in the model. We apply HyperMap to the Autonomous Systems (AS) Internet and find that: 1) the method produces meaningful results, identifying soft communities of ASes belonging to the same geographic region; 2) the method has a remarkable predictive power: Using the resulting map, we can predict missing links in the Internet with high precision, outperforming popular existing methods; and 3) the resulting map is highly navigable, meaning that a vast majority of greedy geometric routing paths are successful and low-stretch. Even though the method is not without limitations, and is open for improvement, it occupies a unique attractive position in the space of tradeoffs between simplicity, accuracy, and computational complexity.