A t-private private information retrieval (PIR) scheme allows a user to retrieve the i-th bit of an n-bit string x replicated among k servers, while any coalition of up to t servers learns no information about i. We present a new geometric approach to PIR, and obtain (1) A t-private k-server protocol with smaller communication complexity, answering an open question of [IK99], (2) A 2-server protocol with n^{1/3} communication, polynomial preprocessing, and online work O(n/log^r n) for any constant r, improving the O(n/log^2 n ) work of Beimel et al [BIM00], and (3) Smaller communication for instance hiding, PIR with a polylogarithmic number of servers, robust PIR, and PIR with fixed answer sizes. To illustrate the power of our approach, we also give alternative, geometric proofs of some of the best 1-private upper bounds of [BIKR02].