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This book deals with the theory of Kac algebras and their dual ity, elaborated independently by M. Enock and J . -M. Schwartz, and by G. . Kac and L. . Vajnermann in the seventies. The sub ject has now reached a state of maturity which fully justifies the publication of this book. Also, in recent times, the topic of "quantum groups" has become very fashionable and attracted the attention of more and more mathematicians and theoret ical physicists. One is still missing a good characterization of quantum groups among Hopf algebras, similar to the character ization of Lie groups among locally...