Explore the cutting-edge intersections of representation theory, homological algebra, and cluster algebras with this comprehensive text. Designed for graduate students and young researchers, this book presents six in-depth mini-courses that delve into the rapidly developing theories connecting these mathematical fields. Gain insights into advanced homological algebra, the representation theory of algebras, cluster algebras arising from surfaces, and more. Originally delivered at a CIMPA research school, this collection offers a valuable foundation for understanding the fruitful interplay between representation theory's categorification of cluster algebras and the new objects of study provided by cluster algebras for representation theory. Perfect for mathematicians seeking to expand their knowledge in this dynamic area. This hardback edition features 223 pages and includes 41 black and white illustrations. It is the first edition, written in English by Ibrahim Assem and Sonia Trepode, and published by Springer Cham. Keywords: Homological Methods, Representation Theory, Cluster Algebras, Algebras, Algebra, Mathematics, Graduate Students, Research, Noncommutative Algebra, Homological Algebra, Finite-dimensional Algebras, CIMPA, Universidad Nacional de Mar del Plata, M.I. Platzeck, Advanced Homological Algebra, Representation Theory of Algebras, Auslander-Reiten Theory, Cluster Algebra Theory, K-theory, Brauer Graph Algebras.