This book presents the fundamentals of vector retrieval. To this end, it delves into important data structures and algorithms that have been successfully used to solve the vector retrieval problem efficiently and effectively. This monograph is divided into four parts. The first part introduces the problem of vector retrieval and formalizes the concepts involved. The second part delves into retrieval algorithms that help solve the vector retrieval problem efficiently and effectively. It includes a chapter each on brand-and-bound algorithms, locality sensitive hashing, graph algorithms, clustering, and sampling. Part three is devoted to vector compression and comprises chapters on quantization and sketching. Finally, the fourth part presents a review of background material in a series of appendices, summarizing relevant concepts from probability, concentration inequalities, and linear algebra. The book emphasizes the theoretical aspects of algorithms and presents related theorems and proofs. It is thus mainly written for researchers and graduate students in theoretical computer science and database and information systems who want to learn about the theoretical foundations of vector retrieval.