This book is intended as the first attempt to gather and unify the results of the theory of noncommutative semigroup rings. Most of the material comes from the literature of the past 10 years, and several new results are included. We follow the line initiated by Infinite Group Rings by D. S. Passman and his later monograph The Algebraic Structure of Group Rings, and R. Gilmer's Commutative Semigroup Rings. The group ring results are basically the starting point for most of the topics considered, while the case of commutative semigroup rings is a base and a motivation for developing the theory of PI -semigroup-algebras. Since problems on semigroup rings R[S] over an arbitrary ring R often are easily settled once we handle the case where R is a field, or they lead to specific difficult problems on tensor products, only semigroup rings with coefficients in a field (referred to as semigroup algebras) will be considered.