Computing shortest paths is a fundamental primitive for several social network applications including socially-sensitive ranking, location-aware search, social auctions and social network privacy. Since these applications compute paths in response to a user query, the goal is to minimize latency while maintaining feasible memory requirements. We present ASAP, a system that achieves this goal by exploiting the structure of social networks. ASAP preprocesses a given network to compute and store a partial shortest path tree (PSPT) for each node. The PSPTs have the property that for any two nodes, each edge along the shortest path is with high probability contained in the PSPT of at least one of the nodes. We show that the structure of social networks enable the PSPT of each node to be an extremely small fraction of the entire network; hence, PSPTs can be stored efficiently and each shortest path can be computed extremely quickly. For a real network with 5 million nodes and 69 million edges, ASAP computes a shortest path for most node pairs in less than 49 microseconds per pair. ASAP, unlike any previous technique, also computes hundreds of paths (along with corresponding distances) between any node pair in less than 100 microseconds. Finally, ASAP admits efficient implementation on distributed programming frameworks like MapReduce.