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This paper is concerned with distributed algorithm for finding shortest paths in an asynchronous communication network. For the problem of Breadth First Search, the best previously known algorithms required either &THgr;(V) time, or &THgr; (E + V · D) communication. We present new algorithm, which requires O(D1+&egr;) time, and O(E1+&egr;) messages, for any &egr; 0. (Here, V is number of nodes, E is number of edges and D is the diameter.) This constitutes a major step towards achieving the lower bounds, which are &OHgr;(E) communication and &OHgr;(D) time.For the general (weighted) shortest paths problem, previously known shortest-paths algorithms required O(k · V2) messages and O(V · logk V) time. Our algorithm requires O(E1 + &egr; · log W) messages and O (V1 + &egr; · log W) time.Our results enable to improve significantly solutions for other basic network problems (e.g. leader election).