This chapter introduces the six families of classical simple groups: linear, unitary, symplectic, and three families of orthogonal groups. The primary goals are to define these groups, prove their simplicity, and describe their automorphisms, subgroups, and covering groups. The chapter begins with basic facts about finite fields and constructs general linear groups and their subquotients $\mathrm{PSL}_n(q)$, which are often simple. Iwasawa’s Lemma is used to prove simplicity, and important subgroups are constructed using the geometry of the underlying vector space. This leads to proofs of several remarkable isomorphisms between $\mathrm{PSL}_n(q)$ and other groups, such as $\mathrm{PSL}_2(4) \cong \mathrm{PSL}_2(5) \cong A_5$ and $\mathrm{PSL}_2(9) \cong A_6$. The other classical groups are defined in terms of certain 'forms' on the vector space. Section 3.4 collects key facts about these forms before introducing the symplectic groups in Section 3.5. The symplectic groups are relatively easy to understand, and their orders, simplicity, subgroups, covering groups, and automorphisms are discussed. The generic isomorphism $\mathrm{Sp}_2(q) \cong \mathrm{SL}_2(q)$ and the exceptional isomorphism $\mathrm{Sp}_4(2) \cong S_6$ are proven. The orthogonal groups present more challenges, especially when the underlying field has characteristic 2. The chapter first treats orthogonal groups over fields of odd order, defining the three types and calculating their orders. The spinor norm is used to obtain simple groups, and their subgroup structure is described. The differences when the field has characteristic 2 are discussed, and the quasideterminant is used to obtain simple groups. Clifford algebras and spin groups are constructed to complete key proofs, such as the well-definition of the spinor norm. A simple version of the Aschbacher–Dynkin theorem is proved, classifying maximal subgroups in classical groups. The generic isomorphisms of orthogonal groups are derived from the Klein correspondence, leading to proofs of exceptional isomorphisms. The chapter concludes with a discussion of exceptional behavior in small classical groups and their relation to exceptional Weyl groups.This chapter introduces the six families of classical simple groups: linear, unitary, symplectic, and three families of orthogonal groups. The primary goals are to define these groups, prove their simplicity, and describe their automorphisms, subgroups, and covering groups. The chapter begins with basic facts about finite fields and constructs general linear groups and their subquotients $\mathrm{PSL}_n(q)$, which are often simple. Iwasawa’s Lemma is used to prove simplicity, and important subgroups are constructed using the geometry of the underlying vector space. This leads to proofs of several remarkable isomorphisms between $\mathrm{PSL}_n(q)$ and other groups, such as $\mathrm{PSL}_2(4) \cong \mathrm{PSL}_2(5) \cong A_5$ and $\mathrm{PSL}_2(9) \cong A_6$. The other classical groups are defined in terms of certain 'forms' on the vector space. Section 3.4 collects key facts about these forms before introducing the symplectic groups in Section 3.5. The symplectic groups are relatively easy to understand, and their orders, simplicity, subgroups, covering groups, and automorphisms are discussed. The generic isomorphism $\mathrm{Sp}_2(q) \cong \mathrm{SL}_2(q)$ and the exceptional isomorphism $\mathrm{Sp}_4(2) \cong S_6$ are proven. The orthogonal groups present more challenges, especially when the underlying field has characteristic 2. The chapter first treats orthogonal groups over fields of odd order, defining the three types and calculating their orders. The spinor norm is used to obtain simple groups, and their subgroup structure is described. The differences when the field has characteristic 2 are discussed, and the quasideterminant is used to obtain simple groups. Clifford algebras and spin groups are constructed to complete key proofs, such as the well-definition of the spinor norm. A simple version of the Aschbacher–Dynkin theorem is proved, classifying maximal subgroups in classical groups. The generic isomorphisms of orthogonal groups are derived from the Klein correspondence, leading to proofs of exceptional isomorphisms. The chapter concludes with a discussion of exceptional behavior in small classical groups and their relation to exceptional Weyl groups.