We study the problem of (\textit{vector set search}) with (\textit{vector set queries}). This task is analogous to traditional near-neighbor search, with the exception that both the query and each element in the collection are (\textit{sets}) of vectors. We identify this problem as a core subroutine for semantic search applications and find that existing solutions are unacceptably slow. Towards this end, we present a new approximate search algorithm, DESSERT (({\bf D})ESSERT ({\bf E})ffeciently ({\bf S})earches ({\bf S})ets of ({\bf E})mbeddings via ({\bf R})etrieval ({\bf T})ables). DESSERT is a general tool with strong theoretical guarantees and excellent empirical performance. When we integrate DESSERT into ColBERT, a state-of-the-art semantic search model, we find a 2-5x speedup on the MS MARCO and LoTTE retrieval benchmarks with minimal loss in recall, underscoring the effectiveness and practical applicability of our proposal.