Krichever and Novikov introduced certain classes of infinite dimensional Lie algebras to extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them to a more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation...