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The problem whether singularities are removable by coordinate transformation is central to General Relativity (GR). I am going to report on a recent breakthrough regarding the question whether there always exists coordinates in which the gravitational metric of GR exhibits optimal regularity (i.e., two derivatives above the Riemann curvature) or whether regularity singularities exist: We introduce a system of elliptic partial differential equations on spacetime (the RT-equations) which determines whether coordinates exist in which the metric exhibits optimal regularity. By developing an existence theory for the (nonlinear) RT-equations, we then prove that optimal metric regularity can always be achieved and that no regularity singularities exist above a threshold level of smoothness. Without resolving the problem of optimal metric regularity the initial value problem of GR remains incomplete. For fluid dynamical shock wave solutions of the Einstein equations, optimal metric regularity is the threshold smoothness that guarantees causal structures, the Newtonian limit to classical Physics, strong (point-wise) solutions and the Hawking-Penrose singularity theorems. Extending the existence theory for the RT-equations to the case of GR shock waves is work in progress. Since the RT-equations are elliptic regardless of metric signature, one may picture the geometry of the gravitational metric to be spanned over the curvature of spacetime like the skin of a drum.