  In this paper we consider fractional Sobolev spaces equipped with weights
being powers of the distance to the boundary of the domain. We prove the
versions of Bourgain--Brezis--Mironescu and Maz'ya--Shaposhnikova asymptotic
formulae for weighted fractional Gagliardo seminorms. For $p>1$ we also provide
a nonlocal characterization of classical weighted Sobolev spaces with power
weights.
