  In this work we analyze a class of nonlinear fractional elliptic systems
involving Hardy--type potentials and coupled by critical Hardy-Sobolev--type
nonlinearities in $\mathbb{R}^N$. Due to the lack of compactness at the
critical exponent the variational approach requires a careful analysis of the
Palais-Smale sequences. In order to overcome this loss of compactness, by means
of a concentration--compactness argument the compactness of PS sequences is
derived. This, combined with a energy characterization of the semi-trivial
solutions, allow us to conclude the existence of positive ground and bound
state solutions en terms of coupling parameter $\nu>0$ and the involved
exponents $\alpha,\beta$.
