**Summary:** This paper presents a comprehensive classification of cluster algebras of finite type, establishing a correspondence between these algebras and the Cartan-Killing classification of semisimple Lie algebras and finite root systems. The authors introduce the concept of cluster algebras through the study of pseudomanifolds and generalized associahedra, providing a combinatorial framework for understanding their structure. Key results include the identification of cluster complexes with dual simplicial complexes of generalized associahedra, and the characterization of finite type cluster algebras via conditions on their exchange matrices. The paper also explores the geometric realizations of cluster algebras, showing how they arise as coordinate rings of algebraic varieties. The main theorem (Theorem 1.4) confirms that cluster algebras of finite type are in one-to-one correspondence with the Cartan-Killing types of semisimple Lie algebras. The proofs rely on deep connections between cluster algebras, root systems, and combinatorial structures such as associahedra, with detailed constructions and properties of cluster variables, mutations, and seed mutations playing a central role. The paper concludes with examples of geometric realizations of cluster algebras of classical types, demonstrating their relevance in algebraic geometry and representation theory.**Summary:** This paper presents a comprehensive classification of cluster algebras of finite type, establishing a correspondence between these algebras and the Cartan-Killing classification of semisimple Lie algebras and finite root systems. The authors introduce the concept of cluster algebras through the study of pseudomanifolds and generalized associahedra, providing a combinatorial framework for understanding their structure. Key results include the identification of cluster complexes with dual simplicial complexes of generalized associahedra, and the characterization of finite type cluster algebras via conditions on their exchange matrices. The paper also explores the geometric realizations of cluster algebras, showing how they arise as coordinate rings of algebraic varieties. The main theorem (Theorem 1.4) confirms that cluster algebras of finite type are in one-to-one correspondence with the Cartan-Killing types of semisimple Lie algebras. The proofs rely on deep connections between cluster algebras, root systems, and combinatorial structures such as associahedra, with detailed constructions and properties of cluster variables, mutations, and seed mutations playing a central role. The paper concludes with examples of geometric realizations of cluster algebras of classical types, demonstrating their relevance in algebraic geometry and representation theory.