Multilayer buffer layers containing BaZrO3 thin films on the bottom and SrTiO3 thin films on the top were successfully applied to improve the epitaxial growth of the YBa2Cu3O7−x (YBCO) films on the MgO substrates, which have the best correspondence of the thermal expansion coefficient with one of YBCO. The HRTEM and X-ray diffraction studies demonstrated pure c-axis orientation and absence of the in-plane misoriented grains in the YBCO films grown on the buffered MgO substrates. High superconducting parameters and no cracks were observed in such YBCO films even at their thicknesses ≫1μm. The multilayer flux transformers having total thickness ∼2-μm were prepared on the buffered MgO substrates and demonstrated an improved insulation between the superconducting layers and an increased dynamic range compared to flux transformers on SrTiO3 substrates.