This paper proposes robust inference methods for difference-in-differences and event-study designs that do not require that the parallel trends assumption holds exactly. Instead, the researcher must only impose restrictions on possible violations of the parallel trends assumption, and our methodology then guarantees uniformly valid ("honest") inference when the imposed restrictions are satisfied. We first show that fixed length confidence intervals have near-optimal expected length for a practically-relevant class of restrictions. We next introduce a novel inference procedure that accommodates a wider range of restrictions on the possible violations of parallel trends, and is based on the observation that inference in our setting is equivalent to testing a system of moment inequalities with a large number of linear nuisance parameters. The resulting confidence sets are consistent, and have optimal local asymptotic power for many parameter configurations. We recommend researchers conduct sensitivity analyses to show what conclusions can be drawn under various restrictions on the set of possible violations of parallel trends.