This tutorial provides a comprehensive overview of filtered vector search (fvs). Fvs queries combine vector search with relational operators. The tutorial explores the challenges of integrating vector search into database engines and emphasizes the need for new optimization techniques. It explains the three primary filtered search methods for fvs queries over generic tree-based and graph-based indices and examines the factors influencing the selection of the most efficient method. A key objective is to highlight the importance of achieving stable recall, ideally in a declarative manner, ensuring consistent recall across queries. The tutorial then discusses recent filter-optimized vector indices and concludes by identifying open research challenges in the field of fvs, aiming to inspire further research and development.