In the last years, a lot of work has been devoted to PDEs involving
fractional diffusion. As a consequence of a work of Graham and Zworski,
one can define on a compact manifold conformally covariant operators which
are of fractional order. Naturally associated to these operators is a
notion of curvature, generalizing scalar or Paneitz curvature. We will
review on some recent results concerning prescribing constant fractional
curvature in the regular and singular case.