  In this paper we prove that every proper Lie groupoid admits a
desingularization to a regular proper Lie groupoid. When equipped with a
Riemannian metric, we show that it admits a desingularization to a regular
Riemannian proper Lie groupoid, arbitrarily close to the original one in the
Gromov-Hausdorff distance between the quotient spaces. We construct the
desingularization via a successive blow-up construction on a proper Lie
groupoid. We also prove that our construction of the desingularization is
invariant under Morita equivalence of groupoids, showing that it is a
desingularization of the underlying differentiable stack.
