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These lecture notes provide a self-contained introduction to regularity theory for elliptic equations and systems in divergence form. After a short review of some classical results on everywhere regularity for scalar-valued weak solutions, the presentation focuses on vector-valued weak solutions to a system of several coupled equations. In the vectorial case, weak solutions may have discontinuities and so are expected, in general, to be regular only outside of a set of measure zero. Several methods are presented concerning the proof of such partial regularity results, and optimal regularity is discussed. Finally, a short overview is given on the current state of the art concerning the size of the singular set on which discontinuities may occur.

The notes are intended for graduate and postgraduate students with a solid background in functional analysis and some familiarity with partial differential equations; they will also be of interest to researchers working on related topics.

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Preliminaries.- Introduction to the Setting.- The Scalar Case.- Foundations for the Vectorial Case.- Partial Regularity Results for Quasilinear Systems.

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"The whole text is equipped with many useful and interesting remarks, which helps make the lecture notes very readable. The book seems to be a solid contribution to understanding the problems of the regularity theory." (Eugen Viszus, Mathematical Reviews, March, 2017)

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