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In the recent decades, it became more and more popular for engineers,physicists, and mathematicians alike to put the Maxwell equations into agenerally covariant form. This is particularly useful for understanding thefundamental structure of electrodynamics (conservation of electric charge andmagnetic flux). Moreover, it is ideally suited for applying it to media withlocal (and mainly linear) response behavior. We try to collect the newknowledge that grew out of this development. We would like to ask theparticipants of EMTS 2016 to inform us of work that we may have overlooked inour review.

Generally Covariant Maxwell Theory for Media with a Local Response: Progress since 2000