  This overview is devoted to splitting methods, a class of numerical
integrators intended for differential equations that can be subdivided into
different problems easier to solve than the original system. Closely connected
with this class of integrators are composition methods, in which one or several
low-order schemes are composed to construct higher-order numerical
approximations to the exact solution. We analyze in detail the order conditions
that have to be satisfied by these classes of methods to achieve a given order,
and provide some insight about their qualitative properties in connection with
geometric numerical integration and the treatment of highly oscillatory
problems. Since splitting methods have received considerable attention in the
realm of partial differential equations, we also cover this subject in the
present survey, with special attention to parabolic equations and their
problems. An exhaustive list of methods of different orders is collected and
tested on simple examples. Finally, some applications of splitting methods in
different areas, ranging from celestial mechanics to statistics, are also
provided.
