The theory of partial regular regularity for elliptic systems replaces the classical De Giorgi-Nash-Moser one for scalar equations asserting that solutions are regular outside a negligible closed subset called the singular set. Eventually, Hausdorff dimension estimates on such a set can be given. The singular set is in general non-empty. The theory is classical, started by Giusti & Miranda and Morrey, in turn relying on De Giorgi's seminal ideas for minimal surfaces. I shall present a few results aimed at extending the classical, local partial regularity theory to nonlinear integrodifferential systems and to provide a few basic, general tools in order to prove so called epsilon-regularity theorems in general non-local settings. From recent, joint work with Cristiana De Filippis (Parma) and Simon Nowak (Bielefeld).