Dr. Yuri Prokhorov, the author of this book, is an expert in birational geometry in the field of algebraic geometry. This book is the first significant expository lecture for "complements"; this notion was introduced by Vyacheslav Shokurov quite recently and is important in understanding singularities of a pair consisting of an algebraic variety and a divisor on it. There is currently much ongoing research on this subject, a very active area in algebraic geometry. This book helps the reader to understand the "complement" concept and provides the basic knowledge about the singularities of a pair. The author gives a simple proof of the boundedness of the complements for two dimensional pairs under some restrictive condition, where this boundedness has been conjectured by Shokurov for every dimension. This book contains information and encouragement necessary to attack the problem of the higher dimensional case.