The book is a comprehensive survey of algorithms for computation in groups. It contains the present state of art in most branches of computational group theory, with a strong emphasis on computation in finite permutation groups and in finitely presented groups. It covers also algorithmic aspects of the theory of representations, computation in polycyclic groups, use of rewriting systems in groups, automatic groups and others. One chapter is devoted to existing libraries and databases (such as the atlas of finite groups). The authors emphasise algorithms fast in practice, rather than asymptotically best ones. A detailed discussion of complexity of presented algorithms is beyond the scope of the book but references are provided. The introduction contains a list of most powerful recent and past computer systems for computing in groups. The book is accessible to graduate students and it can be recommended to all mathematicians interested in this field, both as a textbook and as an extensive reference book.