This book is based on lectures of the author on the subject and it keeps the style of oral lectures in a nice and pleasant manner. The book is divided into three parts. Basic notions (Hopf algebras, dual pairings, actions and coactions, the quantum plane, quasitriangular Hopf algebras and its ribbon version, the quantum double), together with a well chosen set of basic examples used throughout the book are presented in the first part. Representation theory and its famous applications in knot theory are treated in the second part using the point of view of braided categories (including (co)module categories, q-Hecke algebras, quantum dimension, algebras in monoidal categories, braided groups and braided differentiation). Applications of these methods to ordinary Hopf algebras are described in the last part of the book. The reader finds here a discussion of (double) bosonisation, Serre relations, R-matrix methods, Hopf algebra factorizations and Lie bialgebras, together with applications to Poisson geometry. Three problems sets can be found at the end of the book. The book is self-contained and supposes only a basic knowledge of algebra and linear algebra. Many intuitive comments and informal remarks, a well chosen set of main examples used systematically in the book and a clear and understandable style make the book very comfortable and useful for students as well as for mathematicians from other fields.

Reviewer:

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