This article, authored by Pierre Gabriel, is a comprehensive study of abelian categories and their applications to module theory. The paper begins with an introduction that provides an overview of the significance of abelian categories, which were introduced by Buchsbaum and Grothendieck to generalize homological methods. The author emphasizes the importance of these categories in algebraic geometry and commutative algebra. The first chapter serves as a review of category theory, covering fundamental concepts such as universes, morphisms, and functors. It also introduces the notion of equivalence between categories, which is crucial for understanding the structure of abelian categories. Chapter II delves into the properties of abelian categories, focusing on left exact functors and injective envelopes. It discusses the localization in abelian categories, including quotient categories and the properties of section functors. Chapter III explores locally Noetherian categories, defining the Krull dimension and studying the structure of injective objects. It also examines pseudo-compact modules and the duality between locally finite categories and pseudo-compact modules. Chapter IV applies the theory to module theory, detailing the structure of modules over Noetherian rings and the construction of completions. It also discusses the theory of quasi-coherent sheaves on Noetherian schemes. Chapter V applies the results to module theory, providing explicit constructions and demonstrating the equivalence between the study of certain quotient categories and the study of completions of modules for specific topologies. Chapter VI examines the application to the study of quasi-coherent sheaves, including the gluing of abelian categories and the properties of sheafification. The paper concludes with a bibliography and a detailed table of contents, providing a comprehensive reference for further study. The author also notes that the work has been presented in three seminars, highlighting its significance in the field of mathematics.This article, authored by Pierre Gabriel, is a comprehensive study of abelian categories and their applications to module theory. The paper begins with an introduction that provides an overview of the significance of abelian categories, which were introduced by Buchsbaum and Grothendieck to generalize homological methods. The author emphasizes the importance of these categories in algebraic geometry and commutative algebra. The first chapter serves as a review of category theory, covering fundamental concepts such as universes, morphisms, and functors. It also introduces the notion of equivalence between categories, which is crucial for understanding the structure of abelian categories. Chapter II delves into the properties of abelian categories, focusing on left exact functors and injective envelopes. It discusses the localization in abelian categories, including quotient categories and the properties of section functors. Chapter III explores locally Noetherian categories, defining the Krull dimension and studying the structure of injective objects. It also examines pseudo-compact modules and the duality between locally finite categories and pseudo-compact modules. Chapter IV applies the theory to module theory, detailing the structure of modules over Noetherian rings and the construction of completions. It also discusses the theory of quasi-coherent sheaves on Noetherian schemes. Chapter V applies the results to module theory, providing explicit constructions and demonstrating the equivalence between the study of certain quotient categories and the study of completions of modules for specific topologies. Chapter VI examines the application to the study of quasi-coherent sheaves, including the gluing of abelian categories and the properties of sheafification. The paper concludes with a bibliography and a detailed table of contents, providing a comprehensive reference for further study. The author also notes that the work has been presented in three seminars, highlighting its significance in the field of mathematics.