  We study conformal harmonic coordinates on Riemannian manifolds. These are
coordinates constructed as quotients of solutions to the conformal Laplace
equation. We show their existence under general conditions. We find that
conformal harmonic coordinates are a close conformal analogue of harmonic
coordinates. We prove up to boundary regularity results for conformal mappings.
We show that Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors
become elliptic operators in conformal harmonic coordinates if one normalizes
the determinant of the metric. We give a corresponding elliptic regularity
result, which includes an analytic case. We prove a unique continuation result
for local conformal flatness for Bach and obstruction flat manifolds. We
discuss and prove existence of conformal harmonic coordinates on Lorentzian
manifolds. We prove unique continuation results for conformal mappings both on
Riemannian and Lorentzian manifolds.
