Embedding-based vector search underpins many important applications, such as recommendation and retrieval-augmented generation (RAG). It relies on vector indices to enable efficient search. However, these indices require storing high-dimensional embeddings and large index metadata, whose total size can be several times larger than the original data (e.g., text chunks). Such high storage overhead makes it difficult, or even impractical, to deploy vector search on personal devices or large-scale datasets. To tackle this problem, we propose LEANN, a storage-efficient index for vector search that recomputes embeddings on the fly instead of storing them, and compresses state-of-the-art proximity graph indices while preserving search accuracy. LEANN delivers high-quality vector search while using only a fraction of the storage (e.g., 5% of the original data) and supporting storage-efficient index construction and updates. On real-world benchmarks, LEANN reduces index size by up to 50x compared with conventional indices, while maintaining SOTA accuracy and comparable latency for RAG applications.