We consider minimizers of nonlocal functionals, like the fractional perimeter, or the fractional anisotropic perimeter, in low dimensions. It is known that a minimizer for the nonlocal perimeter in \R^2 is necessarily an halfplane. We give a quantitative version of this result, in the following sense: we prove that minimizers in a ball of radius $R$ are nearly flat in $B_1$, when $R$ is large enough. Moreover we establish BV estimates and energy estimates in every dimension for the more general notion of stable critical sets. This is a joint work with Joaquim Serra and Enrico Valdinoci.