In this expository lecture we will discuss some recent results concerning fractional Poincaré and Poincaré-Sobolev inequalities with weights, the degeneracy. These results improve some well known estimates due to Fabes-Kenig-Serapioni from the 80’s in connection with the local regularity of solutions of degenerate elliptic equations and also some more recent results by Bourgain-Brezis-Minorescu. Our approach is different from the usual ones and is based on methods that come from Harmonic Analysis. This is especially visible when the connection of this theory with the BMO space and its different variants will be shown.