  B\"acklund transformations (BTs) are traditionally regarded as a tool for
integrating nonlinear partial differential equations (PDEs). Their use has been
recently extended, however, to problems such as the construction of recursion
operators for symmetries of PDEs, as well as the solution of linear systems of
PDEs. In this article, the concept and some applications of BTs are reviewed.
As an example of an integrable linear system of PDEs, the Maxwell equations of
electromagnetism are shown to constitute a BT connecting the wave equations for
the electric and the magnetic field; plane-wave solutions of the Maxwell system
are constructed in detail. The connection between BTs and recursion operators
is also discussed.
