There is a lack of effective methods for studying properties of finitely generated commutative monoids. This was one of the main reasons for developing a self-contained book on finitely generated commutative monoids with the theory and algorithms needed for the study of the main classical problems related to this kind of monoid. This book is not only addressed to people working in Semigroup Theory. The only knowledge required to follow and understand its contents is basic Linear Algebra. This monograph can also be used as a textbook of a course on finitely generated monoids.