  In this survey we discuss a wide variety of aspects related to Lie group
integrators. These numerical integration schemes for differential equations on
manifolds have been studied in a general and systematic manner since the 1990s
and the activity has since then branched out in several different subareas,
focussing both on theoretical and practical issues. From two alternative
setups, using either frames or Lie group actions on a manifold, we here
introduce the most important classes of schemes used to integrate nonlinear
ordinary differential equations on Lie groups and manifolds. We describe a
number of different applications where there is a natural action by a Lie group
on a manifold such that our integrators can be implemented. An issue which is
not well understood is the role of isotropy and how it affects the behaviour of
the numerical methods. The order theory of numerical Lie group integrators has
become an advanced subtopic in its own right, and here we give a brief
introduction on a somewhat elementary level. Finally, we shall discuss Lie
group integrators having the property that they preserve a symplectic structure
or a first integral.
