This monograph shows how homological algebra helps one to connect, understand more deeply and prove important results in commutative algebra, representation theory and combinatorics. The key tool used here is the Koszul complex and its generalizations. The book is divided into seven chapters: after a preliminary chapter 1, the second chapter deals with local ring theory and proves Serre's characterization of regular local rings and their factoriality. Chapter 3 introduces generalized Koszul complexes and presents some of their recent applications. In chapter 4 structure theorems for finite free resolutions (extending the Hilbert-Burch theorem) are proved. Chapters 5-6 and the appendix concentrate on determinantal ideals and characteristic-free representation theory. However, they also involve a good deal of combinatorics, especially when dealing with the structure of Weyl and Schur modules and their applications (via tableaux and letter place algebras). By carefully selecting the material and clearly presenting the key ideas, the authors have succeeded in writing a very nice and useful monograph both for graduate students and for experts in algebra and representation theory.