The theories of quadratic forms and their applications appear in many parts of mathematics and the sciences. All students of mathematics have the opportunity to encounter such concepts and applications in their first course in linear algebra. This subject and its extensions to infinite dimensions comprise the theory of the numerical range W(T) This book is dedicated to presenting the fundamentals of the subject. In order to make the book quickly informative, the authors place and keep the subject in a complex Hilbert space. This setting is the heart of the numerical range theory for bounded linear operators T and naturally contains the field of values for finite dimensional matrices T. . This book is a particularly well-prepared account of a beautiful and interesting part of functional analysis that also has important applications in numerical analysis.