This book is devoted to the study of scalar and asymptotic scalar derivatives and their applications to the study of some problems considered in nonlinear analysis, in geometry, and in applied mathematics. The notion of a scalar derivative is due to S. Z. Nemeth, ´ and the notion of an asymptotic scalar derivative is due to G. Isac. Both notions are recent, never considered in a book, and have interesting applications. About applications, we cite applications to the study of complementarity problems, to the study of xed points of nonlinear mappings, to spectral nonlinear analysis, and to the study of some interesting problems considered in differential geometry and other applications. A new characterization of monotonicity of nonlinear mappings is another remarkable application of scalar derivatives. A relation between scalar derivatives and asymptotic scalar derivatives, - alized by an inversion operator is also presented in this book. This relation has important consequences in the theory of scalar derivatives, and in some applications. For example, this relation permitted us a new development of the method of exceptional family of elements, introduced and used by G. Isac in complementarity theory. Now, we present a brief description of the contents of this book. Chapter 1 is dedicated to the study of scalar derivatives in Euclidean spaces.