  The limits of linear electrodynamics are reviewed, and possible directions of
nonlinear extension are explored. The central theme is that the qualitative
character of the empirical successes of quantum electrodynamics must be used as
a guide for understanding the nature of the nonlinearity of electrodynamics at
the subatomic level. Some established theories of nonlinear electrodynamics,
namely, those of Mie, Born, and Infeld are presented in the language of the
modern geometrical and topological methods of mathematical physics. The manner
by which spacetime curvature and topology can affect electromagnetism is also
reviewed. Finally, the phenomena of nonlinear optics are discussed as a
possible guide to building one's intuition regarding the process of extending
electrodynamics into nonlinearity in a manner that is consistent with the
qualitative and empirical results of quantum electrodynamics.
