We establish trace Hardy and trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space our results cover the full range of the exponent s\in (0,1) of the fractional Laplacians. Thus we answer in particular an open problem raised by Frank and Seiringer, which has been also answered but only in the range of s\in (1/2,1) and through different approaches by Dyda in 2010, Sloane in 2010 and by Dyda and Frank in 2011. The results presented in this talk are contained in two different articles in collaboration with S. Filippas and A. Tertikas, the second one only recently submitted.