ANN (Approximate Nearest Neighbors) algorithms trade exact accuracy for dramatically faster search in high-dimensional spaces. The problem: Exact k-NN requires comparing query to all vectors - O(N) per query. Infeasible at scale. Approximation: Accept that returned neighbors might not be the true k closest, but are very close. Much faster. Quality measure: Recall@k - fraction of true k nearest neighbors found by approximate method. 95%+ common target. Major approaches: Graph-based (HNSW): Navigate proximity graph. Best accuracy/speed. Tree-based (Annoy): Random projection trees. Simple, efficient. Hashing (LSH): Locality-sensitive hashing. Probabilistic. Clustering (IVF): Partition space, search relevant clusters. Quantization (PQ): Compress vectors, approximate distances. Speed gains: Milliseconds instead of seconds for million-vector search. Orders of magnitude improvement. Trade-offs: Each method has different accuracy/speed/memory characteristics. Index building time varies. Tuning: Parameters control accuracy vs speed. More compute at query time = higher recall. Choice depends on: Dataset size, dimensionality, memory constraints, latency requirements, update frequency. Foundation for modern vector search systems.