Around 1978, a mutation of associative algebras was introduced to generalize the formalism of classical mechanics as well as quantum mechanics. This volume presents a self-contained and detailed treatment of the mathematical theory of mutation algebras, which is based on research in this subject since 1989. The book also deals with a broader class of algebras, mutations of alternative algebras, which are a natural generalization of mutations of associative algebras. A complete structure theory, including automorphisms, derviations and certain representations, is given for mutations of artinian alternative algebras, and, in particular, of Cayley-Dickson algebras. Since the mutation algebras do not form a variety, the structure theory explored in this volume takes quite a different approach from the standard theory of nonassociative algebras and provides an important interplay with the theory of noncommutative (associative) algebras through mutation parameters. Simple algebras and open problems presented in this book should stimulate additional research and applications in the area. This book should be of value to graduate students, mathematicians and physicists interested in applications of algebras.