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The aim of this paper is to report on recent development on the conformal fractional Laplacian, both from the analytic and geometric points of view, but especially towards the PDE community.

The basic tool in conformal geometry is to understand the geometry of a space by studying the transformations on such space. More precisely, let Mn be a smooth Riemannian manifold of dimension n ě 2 with a Riemannian metric h. A conformal change of metric is such that it preserves angles so, mathematically, two metrics h, h are conformally related if