Many differential equations of practical interest evolve on Lie groups or on
manifolds acted upon by Lie groups. The retention of Lie-group structure
under discretization is often vital in the recovery of qualitatively correct geometry
and dynamics and in the minimization of numerical error. Having
introduced requisite elements of differential geometry, this paper surveys the
novel theory of numerical integrators that respect Lie-group structure, highlighting
theory, algorithmic issues and a number of applications.