The chapter "Fundamentals of Semigroup Theory" by John M. Howie, a Regius Professor of Mathematics at the University of St Andrews, provides a comprehensive introduction to the theory of semigroups. It covers fundamental concepts and structures, including basic definitions, monogenic semigroups, ordered sets, semilattices, lattices, binary relations, equivalences, congruences, free semigroups and monoids, ideals, Rees congruences, and lattices of equivalences and congruences. The chapter also delves into Green's equivalences and regular semigroups, discussing $D$-classes, regular $D$-classes, regular semigroups, and the sandwich set. Additionally, it explores 0-simple semigroups, simple and 0-simple semigroups, principal factors, the Rees Theorem, completely simple semigroups, isomorphism and normalization, congruences on completely 0-simple semigroups, finite congruence-free semigroups, and exercises. The text further examines completely regular semigroups, Clifford decompositions, Clifford semigroups, varieties, bands, and free bands. It also covers inverse semigroups, including preliminaries, the natural order relation, congruences, the Munn semigroup, anti-uniform semilattices, bisimple inverse semigroups, simple inverse semigroups, representations, $E$-unitary inverse semigroups, and free inverse monoids. The chapter concludes with sections on other classes of regular semigroups, such as locally inverse semigroups, orthodox semigroups, and semibands, along with exercises and notes.The chapter "Fundamentals of Semigroup Theory" by John M. Howie, a Regius Professor of Mathematics at the University of St Andrews, provides a comprehensive introduction to the theory of semigroups. It covers fundamental concepts and structures, including basic definitions, monogenic semigroups, ordered sets, semilattices, lattices, binary relations, equivalences, congruences, free semigroups and monoids, ideals, Rees congruences, and lattices of equivalences and congruences. The chapter also delves into Green's equivalences and regular semigroups, discussing $D$-classes, regular $D$-classes, regular semigroups, and the sandwich set. Additionally, it explores 0-simple semigroups, simple and 0-simple semigroups, principal factors, the Rees Theorem, completely simple semigroups, isomorphism and normalization, congruences on completely 0-simple semigroups, finite congruence-free semigroups, and exercises. The text further examines completely regular semigroups, Clifford decompositions, Clifford semigroups, varieties, bands, and free bands. It also covers inverse semigroups, including preliminaries, the natural order relation, congruences, the Munn semigroup, anti-uniform semilattices, bisimple inverse semigroups, simple inverse semigroups, representations, $E$-unitary inverse semigroups, and free inverse monoids. The chapter concludes with sections on other classes of regular semigroups, such as locally inverse semigroups, orthodox semigroups, and semibands, along with exercises and notes.