Modern AI and vector search are rapidly converging, forming a promising research frontier in intelligent information systems. On one hand, advances in AI have substantially improved the semantic accuracy and efficiency of vector search, including learned indexing structures, adaptive pruning strategies, and automated parameter tuning. On the other hand, powerful vector search techniques have enabled new AI paradigms, notably Retrieval-Augmented Generation (RAG), which effectively mitigates challenges in Large Language Models (LLMs) like knowledge staleness and hallucinations. This mutual reinforcement establishes a virtuous cycle where AI injects intelligence and adaptive optimization into vector search, while vector search, in turn, expands AI's capabilities in knowledge integration and context-aware generation. This tutorial provides a comprehensive overview of recent research and advancements at this intersection. We begin by discussing the foundational background and motivations for integrating vector search and AI. Subsequently, we explore how AI empowers vector search (AI4VS) across each step of the vector search pipeline. We then investigate how vector search empowers AI (VS4AI), with a particular focus on RAG frameworks that integrate dynamic, external knowledge sources into the generative process of LLMs. Furthermore, we analyze end-to-end co-optimization strategies that fully unlock the potential of the ``virtuous cycle" between vector search and AI. Finally, we highlight key challenges and future research opportunities in this emerging area. This paper was published in ICDE 2026.