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The problem of finding locally dense components of a graph is an important primitive in data analysis, with wide-ranging applications from community mining to spam detection and the discovery of biological network modules. In this paper we present new algorithms for finding the densest subgraph in the streaming model. For any epsilon>0, our algorithms make O((log n)/log (1+epsilon)) passes over the input and find a subgraph whose density is guaranteed to be within a factor 2(1+epsilon) of the optimum. Our algorithms are also easily parallelizable and we illustrate this by realizing them in the MapReduce model. In addition we perform extensive experimental evaluation on massive real-world graphs showing the performance and scalability of our algorithms in practice.