  We show global uniqueness in an inverse problem for the fractional
Schr\"odinger equation: an unknown potential in a bounded domain is uniquely
determined by exterior measurements of solutions. We also show global
uniqueness in the partial data problem where the measurements are taken in
arbitrary open, possibly disjoint, subsets of the exterior. The results apply
in any dimension $\geq 2$ and are based on a strong approximation property of
the fractional equation that extends earlier work. This special feature of the
nonlocal equation renders the analysis of related inverse problems radically
different from the traditional Calder\'on problem.
