This chapter describes the six families of classical simple groups: linear, unitary, symplectic, and three families of orthogonal groups. These groups are derived from matrix groups by taking the quotient by their centers. The chapter aims to define these groups, prove their simplicity, and describe their automorphisms, subgroups, and covering groups. It begins with finite fields, then constructs general linear groups and their subquotients, proving their simplicity using Iwasawa's Lemma. It discusses projective spaces and isomorphisms between certain groups like PSL₂(4) ≅ PSL₂(5) ≅ A₅ and PSL₂(9) ≅ A₆. Covering groups and automorphism groups are briefly mentioned. The symplectic groups are defined using forms on vector spaces, and their orders, simplicity, subgroups, and automorphisms are discussed. The unitary groups are treated similarly. The orthogonal groups present the most difficulties, especially in fields of characteristic 2. The spinor norm is used to obtain simple groups, and the quasideterminant is used in characteristic 2. Clifford algebras and spin groups are introduced to prove key facts, including the well-definedness of the spinor norm. A simple version of the Aschbacher–Dynkin theorem is proved, classifying maximal subgroups in classical groups. This theorem is analogous to the O'Nan–Scott theorem for permutation groups but is harder to prove and less explicit. The generic isomorphisms of orthogonal groups are derived from the Klein correspondence, leading to isomorphisms like PΩ₆⁺(q) ≅ PSL₄(q). The chapter also discusses exceptional isomorphisms and the behavior of small classical groups, relating them to exceptional Weyl groups. Finite fields are defined, with the multiplicative group of non-zero elements being cyclic. For every prime p and positive integer d, there exists a unique field of order p^d up to isomorphism.This chapter describes the six families of classical simple groups: linear, unitary, symplectic, and three families of orthogonal groups. These groups are derived from matrix groups by taking the quotient by their centers. The chapter aims to define these groups, prove their simplicity, and describe their automorphisms, subgroups, and covering groups. It begins with finite fields, then constructs general linear groups and their subquotients, proving their simplicity using Iwasawa's Lemma. It discusses projective spaces and isomorphisms between certain groups like PSL₂(4) ≅ PSL₂(5) ≅ A₅ and PSL₂(9) ≅ A₆. Covering groups and automorphism groups are briefly mentioned. The symplectic groups are defined using forms on vector spaces, and their orders, simplicity, subgroups, and automorphisms are discussed. The unitary groups are treated similarly. The orthogonal groups present the most difficulties, especially in fields of characteristic 2. The spinor norm is used to obtain simple groups, and the quasideterminant is used in characteristic 2. Clifford algebras and spin groups are introduced to prove key facts, including the well-definedness of the spinor norm. A simple version of the Aschbacher–Dynkin theorem is proved, classifying maximal subgroups in classical groups. This theorem is analogous to the O'Nan–Scott theorem for permutation groups but is harder to prove and less explicit. The generic isomorphisms of orthogonal groups are derived from the Klein correspondence, leading to isomorphisms like PΩ₆⁺(q) ≅ PSL₄(q). The chapter also discusses exceptional isomorphisms and the behavior of small classical groups, relating them to exceptional Weyl groups. Finite fields are defined, with the multiplicative group of non-zero elements being cyclic. For every prime p and positive integer d, there exists a unique field of order p^d up to isomorphism.