<P>Multiplicative invariant theory, as a research area in its own right within the wider spectrum of invariant theory, is of relatively recent vintage. The present text offers a coherent account of the basic results achieved thus far..</P><P>Multiplicative invariant theory is intimately tied to integral representations of finite groups. Therefore, the field has a predominantly discrete, algebraic flavor. Geometry, specifically the theory of algebraic groups, enters through Weyl groups and their root lattices as well as via character lattices of algebraic tori.</P><P>Throughout the text, numerous explicit examples of multiplicative invariant algebras and fields are presented, including the complete list of all multiplicative invariant algebras for lattices of rank 2.</P><P>The book is intended for graduate and postgraduate students as well as researchers in integral representation theory, commutative algebra and, mostly, invariant theory.</P>
