We consider dynamic selection problems, where a decision maker repeatedly selects a set of items from a larger collection of available items. A classic example is the dynamic assortment problem with demand learning, where a retailer chooses items to offer for sale subject to a display space constraint. The retailer may adjust the assortment over time in response to the observed demand. These dynamic selection problems are naturally formulated as stochastic dynamic programs (DPs) but are difficult to solve because the optimal selection decisions depend on the states of all items. In this paper, we study heuristic policies for dynamic selection problems and provide upper bounds on the performance of an optimal policy that can be used to assess the performance of a heuristic policy. The policies and bounds that we consider are based on a Lagrangian relaxation of the DP that relaxes the constraint limiting the number of items that may be selected. We characterize the performance of the Lagrangian index policy and bound and show that, under mild conditions, these policies and bounds are asymptotically optimal for problems with many items; mixed policies and tiebreaking play an essential role in the analysis of these index policies and can have a surprising impact on performance. We demonstrate these policies and bounds in two large scale examples: a dynamic assortment problem with demand learning and an applicant screening problem.