Here is the first pedagogical introduction to harmonic superspace in extended supersymmetry by the scientists who developed the method. Inspired by developments in superstring theory, it provides a systematic treatment of the quantum field theories with N=2 and N=3 supersymmetry in Harmonic Superspace. The authors present the harmonic superspace approach as a means of providing an off-shell description of the N=2 supersymmetric theories, both at the classical and quantum levels. Furthermore, they show how it offers a unique way to construct an off-shell formulation of a theory with higher supersymmetry, namely, the N=3 supersymmetric Yang-Mills theory. Harmonic Superspace makes manifest many remarkable geometric properties of the N=2 theories, for example, the one-to-one correspondence between N=2 supersymmetric matter, and hyper-Kähler and quaternionic manifolds. "...this book, written by the group which invented the method, contains a great deal of useful information and is a welcome survey of the early results starting from the pioneering paper of Galperin et al." Mathematical Reviews "...the first discussion, outside of the confines of the research literature, to present pedagogically the intricacies of its topic. This is a unique book, a readable narrative that clearly explains the applications and achievements of the approach. The proseois clear, with a minimum of jargon. It succeeds in its goals" Physics Today The inventors of harmonic superspace present a clear account of the theory and its applications.