This book, consisting of two volumes, gives a contemporary account of the study of the class of projective algebraic surfaces known as Enriques surfaces. These surfaces were discovered more than 125 years by F. Enriques in an attempt to extend the characterization of rational algebraic curves to the case of algebraic surfaces. The novel feature of the present exposition is that no assumption on the characteristic of the ground field is assumed. This requirement calls for exploring the geometry of such surfaces by purely geometric and arithmetic methods that do not rely on transcendental methods such as the theory of periods of algebraic surfaces of type K3, which are close relatives of Enriques surfaces. Some of the methods use many technical tools from algebraic geometry that are discussed in Volume 1 and will be a useful source of reference for the study of algebraic surfaces over fields of positive characteristic. Volume 1 also contains a detailed exposition of the theory of
