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In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number s>2, we prove that every graph on n vertices with average degree d≥s contains a subgraph of average degree at least s on at most nd^-ss-2(logd)^O(s)1 vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with n vertices and average degree at least n^1-2s+ε contains a subgraph of average degree at least s on O_ε,s(1) vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraëte.