Suppose that you have a Riemannian metric in a certain coordinate system in which is described by C^k functions. Are there any other coordinates in which the metric is more regular? In other words, is this regularity due to a geometric property of the manifold or is just a poor choice of coordinates? In the talk, we will address this question and try to make precise why harmonic coordinates are "the best" coordinates. As an application, we will study the analyticity of the metric in Einstein manifolds and, if we have time, an application to the convergence of sequences of manifolds.