  In 1933-1934 Born and Infeld constructed the first non-linear generalization
of Maxwell's electrodynamics that turned out to be a remarkable theory in many
respects. In 1935 Heisenberg and Euler computed a complete effective action
describing non-linear corrections to Maxwell's theory due to quantum
electron-positron one-loop effects. Since then, these and a variety of other
models of non-linear electrodynamics proposed in the course of decades have
been extensively studied and used in a wide range of areas of theoretical
physics including string theory, gravity, cosmology and condensed matter (CMT).
In these notes I will overview general properties of non-linear electrodynamics
and particular models which are distinguished by their symmetries and physical
properties, such as a recently discovered unique non-linear modification of
Maxwell's electrodynamics which is conformal and duality invariant. I will also
sketch how non-linear electromagnetic effects may manifest themselves in
physical phenomena (such as vacuum birefringence), in properties of
gravitational objects (e.g. charged black holes) and in the evolution of the
universe, and can be used, via gravity/CMT holography, for the description of
properties of certain conducting and insulating materials.
