The book describes methods for working with elements, subgroups, and quotient groups of a finitely presented group. The author emphasizes the connection with fundamental algorithms from theoretical computer science, particularly the theory of automata and formal languages, from computational number theory, and from computational commutative algebra. The LLL lattice reduction algorithm and various algorithms for Hermite and Smith normal forms are used to study the Abelian quotients of a finitely presented group. The work of Baumslag, Cannonito, and Miller on computing non-Abelian polycyclic quotients is described as a generalization of Buchberger's Gröbner basis methods to right ideals in the integral group ring of a polycyclic group.