In Riemannian geometry, harmonic coordinates are known to provide the best possible framework when regularity issues of geometric quantities like the metric are relevant to find solutions of PDE's on manifolds. In particular, (geometric) Killing spinors on spin manifolds with metric of Hölder regularity C^{1,alpha} are weak solutions of the (vacuum's) Einstein equation and the metric turns out to be smooth on harmonic coordinate charts. This makes the manifold Einstein in the usual sense. Known results on convergence of metrics in the Riemannian and Lorentzian context will be mentioned.