Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard.In this paper, we design the first constant-factor approximation algorithms for maximizing nonnegative submodular functions. In particular, we give a deterministic local search 1 3 -approximation and a randomized 2 5 -approximation algorithm for maximizing nonnegative submodular functions. We also show that a uniformly random set gives a 1 4 -approximation. For symmetric submodular functions, we show that a random set gives a 1 2 -approximation, which can be also achieved by deterministic local search.These algorithms work in the value oracle model where the submodular function is accessible through a black box returning f (S) for a given set S. We show that in this model, 1 2 -approximation for symmetric submodular functions is the best one can achieve with a subexponential number of queries. For the case where the function is given explicitly (as a sum of nonnegative submodular functions, each depending only on a constant number of elements), we prove that it is NP-hard to achieve a ( 3 4 + )-approximation in the general case (or a ( 5 6 + )-approximation in the symmetric case).