  Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$
its conformal infinity. Our primary aim in this paper is to introduce the
prescribed fractional scalar curvature problem on $M$ and provide solutions
under various geometric conditions on $X$ and $M$. We also obtain the existence
results for the fractional Yamabe problem in the endpoint case, e.g., $n = 3$,
$\gamma = 1/2$ and $M$ is non-umbilic, etc. Every solution we find turns out to
be smooth on $M$.
