  In the present paper we study embedding operators for weighted Sobolev spaces
whose weights satisfy the well-known Muckenhoupt A_p-condition. Sufficient
conditions for boundedness and compactness of the embedding operators are
obtained for smooth domains and domains with boundary singularities. The
proposed method is based on the concept of 'generalized' quasiconformal
homeomorphisms (homeomorphisms with bounded mean distortion.) The choice of the
homeomorphism type depends on the choice of the corresponding weighted Sobolev
space. Such classes of homeomorphisms induce bounded composition operators for
weighted Sobolev spaces. With the help of these homeomorphism classes the
embedding problem for non-smooth domains is reduced to the corresponding
classical embedding problem for smooth domains. Examples of domains with
anisotropic H\"older singularities demonstrate sharpness of our machinery
comparatively with known results.
