We study the boundary correspondence under quasiconformal self-mappings. in the plane and in space. First, considering quasiconformal mappingsfof the upper half-plane onto itself, we give new sufficient conditions for the boundary correspondence λ↱f(λ) to be aC1-homeomorphism. Then for quasicontormal self-mappingsfof the upper half-spaceHnn≥ 3. we show that both of the well-known Carleson conditions as well as the condition of the approximate continuity for the distortion coefficientKfn:(x) imply that is a similarity mapping. The proofs are based on a regularity theorem due to Tciehmüller. Wittich, Belinskii and Lehto in the plane and similar results due to Reshetnyak in space.