The quantum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. It is shown that these algebras have natural integral forms which can be specialized at roots of 1 and yield new objects which include quantum versions of the semi-simple groups over fields of positive characteristic. The theory of quantum groups has led to a new, extremely rigid structure, in which the objects of the theory are provided with canonical bases with rather remarkable properties. The book contains an extensive treatment of the theory of canonical bases in the framework of perverse sheaves. The theory developed in the book includes the case of quantum affine enveloping algebras and, more generally, the quantum analogs of the KacMoody Lie algebras.This book will be of interest to mathematicians working in the representation theory of Lie groups and Lie algebras, knot theorists and to theoretical physicists and graduate students. Since large parts of the book are independent of the theory of perverse sheaves, the book could also be used as a text book.

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