This book reviews recent results on low-dimensional quantum  field theories  and their connection with quantum group  theory and the theory of braided,   balanced tensor  categories. It presents detailed, mathematically            precise  introductions to these subjects and then continues with             new  results. Among the main results are a detailed analysis of  the      representation theory of U (sl ), for q a primitive  root of unity, and a    semi-simple quotient thereof, a  classfication of braided tensor categories  generated by an  object of q-dimension less than two, and an application     of  these results to the theory of sectors in algebraic quantum  field    theory. This clarifies the notion of "quantized  symmetries" in quantum fieldtheory. The reader is expected  to be familiar with basic notions and resultsin algebra.  The book is intended for research                               mathematicians,  mathematical physicists and graduate students.

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