This book provides a comprehensive treatment of the representations and invariants of classical groups. It begins with an introduction to classical groups as linear algebraic groups, covering topics such as linear algebraic groups, Lie algebras, Jordan decomposition, and real forms. The second chapter discusses the basic structure of classical groups, including semisimple and unipotent elements, irreducible representations of SL(2, C), the adjoint representation, and the reductivity of classical groups. The third chapter focuses on algebras and representations, including associative algebras, simple algebras, commutants, characters, group algebras of finite groups, and representations of finite groups. The fourth chapter explores polynomial and tensor invariants, including polynomial invariants, invariants for classical groups, tensor invariants, and applications of the first fundamental theorem. The fifth chapter presents highest weight theory, discussing irreducible representations, applications, and dualities. The book then covers spinors, including Clifford algebras, spin representations, spin groups, and real forms of spin(n, C). The following chapters discuss cohomology and characters, branching laws, tensor representations of GL(V), tensor representations of O(V) and Sp(V), and connections to invariant theory and knot polynomials. The book also includes a detailed discussion of algebraic groups, homogeneous spaces, Borel subgroups, and further properties of real forms. It concludes with a chapter on representations on spaces of regular functions, including multiplicity-free spaces, regular functions on symmetric spaces, and separation of variables for isotropy representations. The appendices provide additional material on algebraic geometry, linear and multilinear algebra, associative algebras and Lie algebras, and manifolds and Lie groups. The book is a valuable resource for researchers and students in mathematics, particularly in the areas of representation theory and invariant theory.This book provides a comprehensive treatment of the representations and invariants of classical groups. It begins with an introduction to classical groups as linear algebraic groups, covering topics such as linear algebraic groups, Lie algebras, Jordan decomposition, and real forms. The second chapter discusses the basic structure of classical groups, including semisimple and unipotent elements, irreducible representations of SL(2, C), the adjoint representation, and the reductivity of classical groups. The third chapter focuses on algebras and representations, including associative algebras, simple algebras, commutants, characters, group algebras of finite groups, and representations of finite groups. The fourth chapter explores polynomial and tensor invariants, including polynomial invariants, invariants for classical groups, tensor invariants, and applications of the first fundamental theorem. The fifth chapter presents highest weight theory, discussing irreducible representations, applications, and dualities. The book then covers spinors, including Clifford algebras, spin representations, spin groups, and real forms of spin(n, C). The following chapters discuss cohomology and characters, branching laws, tensor representations of GL(V), tensor representations of O(V) and Sp(V), and connections to invariant theory and knot polynomials. The book also includes a detailed discussion of algebraic groups, homogeneous spaces, Borel subgroups, and further properties of real forms. It concludes with a chapter on representations on spaces of regular functions, including multiplicity-free spaces, regular functions on symmetric spaces, and separation of variables for isotropy representations. The appendices provide additional material on algebraic geometry, linear and multilinear algebra, associative algebras and Lie algebras, and manifolds and Lie groups. The book is a valuable resource for researchers and students in mathematics, particularly in the areas of representation theory and invariant theory.