Revised second edition. The text covers the material presented for a graduate-level course on quantum groups at Harvard University. The contents cover: Poisoon algebras and quantization, Poisson-Lie groups, coboundary Lie bialgebras, Drinfelds double construction, Belavin-Drinfeld classification, Infinite dimensional Lie bialgebras, Hopf algebras, Quantized universal enveloping algebras, formal groups and h-formal groups, infinite dimensional quantum groups, the quantum double, tensor categories and quasi Hopf-algebras, braided tensor categories, KZ equations and the Drinfeld Category, Quasi-Hpf enveloping algebras, Lie associators, Fiber functors and Tannaka-Driein duality, Quantization of finite Lie bialgebras, Universal constructions, Universal quantization, Dequantization and the equivalence theorem, KZ associator and multiple zeta functions, and Mondoromy of trigonometric KZ equations. Probems are given with each subject and an answer key is included. Table of contents Poisson algebras and quantization Poisson-Lie groups Coboundary Lie bialgebras Drinfeld's double construction Belavin-Drinfeld classification (I) Infinite dimensional Lie bialgebras Belavin-Drinfeld classification (II) Hopf algebras Quantized universal enveloping algebras Formal groups and h-formal groups Infinite dimensional quantum groups The quantum double Tensor categories and quasi-Hopfalgebras Braided tensor categories KZ equations and the Drinfeld Category Quasi-Hopf quantized enveloping algebras Lie associators Fiber functors and Tannaka-Krein duality Quantization of finite dimensional Lie bialgebras Universal constructions Universal quantization Dequantization and the Equalivalence