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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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