The Adaptive Seeding problem is an algorithmic challenge motivated by influence maximization in social networks: One seeks to select among certain accessible nodes in a network, and then select, adaptively, among neighbors of those nodes as they become accessible in order to maximize a global objective function. More generally, adaptive seeding is a stochastic optimization framework where the choices in the first stage affect the realizations in the second stage, over which we aim to optimize.Our main result is a (1 − 1/e) 2 -approximation for the adaptive seeding problem for any monotone submodular function. While adaptive policies are often approximated via non-adaptive policies, our algorithm is based on a novel method we call locally-adaptive policies. These policies combine a non-adaptive global structure, with local adaptive optimizations. This method enables the (1 − 1/e) 2 -approximation for general monotone submodular functions and circumvents some of the impossibilities associated with non-adaptive policies.We also introduce a fundamental problem in submodular optimization that may be of independent interest: given a ground set of elements where every element appears with some small probability, find a set of expected size at most k that has the highest expected value over the realization of the elements. We show a surprising result: there are classes of monotone submodular functions (including coverage) that can be approxi-