In semigroup theory there are certain kinds of band decompositions, which are very useful in the study of the structure of semigroups. There are a number of special semigroup classes in which these decompositions can be used very successfully. This book focuses attention on such classes of semigroups. Some of them are partially discussed in earlier books, but during the last 30 years of the 20th century new semigroup classes have appeared and a fairly large body of material has been published on them. The book provides a systematic review on this subject. The first chapter is an introduction. The remaining chapters are devoted to special semigroup classes. These are Putcha semigroups, commutative semigroups, weakly commutative semigroups, R-Commutative semigroups, conditionally commutative semigroups, RC-commutative semigroups, quasi commutative semigroups, medial semigroups, right commutative semigroups, externally commutative semigroups, E-m semigroups, WE-m semigroups, weakly exponential semigroups, (m,n)-commutative semigroups and n(2)-permutable semigroups.