this monograph treats comprehensively central aspects of string rewriting systems in the form of semi-thue systems. these are so general as to enable the discussion of all the basic notions and questions that arise in arbitrary replacement systems as used in various areas of computer science. the church-rosser property is used in its original meaning and the existence of complete monoid and group presentations is the central point of discussion. decidability problems with their complexity are surveyed and congruential languages including the deterministic context-free nts languages are discussed. the book contains a number of generalizations of results published elsewhere, e.g., the uniqueness of complete string rewriting systems with respect to the underlying order. completely new and unpublished results which serve as an exposition of techniques and new methods are discussed in detail. with the help of semi-thue systems it is shown in which situations the famous knuth-bendix completion method does not terminate and why, and that in general complete replacement systems cannot always be used as algorithms to solve the word problem. it is suggested how these situations can be stated by using a certain control under which the rewriting is to be performed. this monograph is a reference for graduate students and active researchers in theoretical computer science. the reader is led to the forefront of current research in the area of string rewriting and monoid presentations.