We describe new techniques for proving lower bounds on data-structure problems, with the following broad consequences: • the first Ω(lgn) lower bound for any dynamic problem, improving on a bound that had been standing since 1989; • for static data structures, the first separation between linear and polynomial space. Specifically, for some problems that have constant query time when polynomial space is allowed, we can show Ω(lg n/ lg lg n) bounds when the space is O(n · polylog n). Using these techniques, we analyze a variety of central data-structure problems, and obtain improved lower bounds for the following: • the partial-sums problem (a fundamental application of augmented binary search trees); • the predecessor problem (which is equivalent to IP lookup in Internet routers); • dynamic trees and dynamic connectivity; • orthogonal range stabbing; • orthogonal range counting, and orthogonal range reporting; • the partial match problem (searching with wild-cards); • (1 + ε)-approximate near neighbor on the hypercube; • approximate nearest neighbor in the l∞ metric. Our new techniques lead to surprisingly non-technical proofs. For several problems, we obtain simpler proofs for bounds that were already known.