The story of separately holomorphic functions began about 100 years ago. During the second half of the 19th century, it became known that a separately continuous function is not necessarily continuous as a function of all variables. At the beginning of the 20th century, the study of separately holomorphic functions started due to the fundamental work of Osgood and Hartogs. This book provides the first selfcontained and complete presentation of the study of separately holomorphic functions, from its beginnings to current research. Most of the results presented have never been published before in book form. The text is divided into two parts. The first part deals with separately holomorphic functions, "without singularities". The second part addresses the situation of existing singularities. A discussion of the classical results related to separately holomorphic functions leads to the most fundamental result, the classical cross theorem as well as various extensions and generalizations, to more complicated "crosses". Additionally, several applications for other classes of "separately regular" functions are given. A solid background in basic complex analysis is a prerequisite. To make the book self contained, all the results are collected in special introductory chapters and referred to at the beginning of each section. This book is addressed to students and researchers in several complex variables as well as mathematicians and theoretical physicists interested in this area of mathematics. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students, research mathematicians, and theoretical physicists interested in complex variables and analysis. Table of Contents Cross theorems without singularities  Classical results I
 Prerequisites
 Relative extremal functions
 Classical results II
 Classical cross theorem
 Discs method
 Nonclassical cross theorems
 Boundary cross theorems
Cross theorems with singularities  Extension with singularities
 Cross theorem with singularities
 Separately meromorphic functions
 Bibliography
 Symbols
 List of symbols
 Subject index
