Understanding the spatial networks formed by the trajectories of mobile users can be beneficial to applications ranging from epidemiology to local search. Despite the potential for impact in a number of fields, several aspects of human mobility networks remain largely unexplored due to the lack of large-scale data at a fine spatiotemporal resolution. Using a longitudinal dataset from the location-based service Foursquare, we perform an empirical analysis of the topological properties of place networks and note their resemblance to online social networks in terms of heavy-tailed degree distributions, triadic closure mechanisms and the small world property. Unlike social networks however, place networks present a mixture of connectivity trends in terms of assortativity that are surprisingly similar to those of the web graph. We take advantage of additional semantic information to interpret how nodes that take on functional roles such as `travel hub', or `food spot' behave in these networks. Finally, motivated by the large volume of new links appearing in place networks over time, we formulate the classic link prediction problem in this new domain. We propose a novel variant of gravity models that brings together three essential elements of inter-place connectivity in urban environments: network-level interactions, human mobility dynamics, and geographic distance. We evaluate this model and find it outperforms a number of baseline predictors and supervised learning algorithms on a task of predicting new links in a sample of one hundred popular cities.