An approach to high quality heteroepitaxial Ge growth on Si substrates by MBE has been studied. In the method studied the Ge layers were grown by a two-step procedure involving a thermal annealing process between the first and second Ge layer growth. It was found that the surface morphology of the Ge depended on the thickness of the first layer, and its thickness of more than 100 nm resulted in smooth surfaces. The crystalline quality of the Ge was affected by both the first and second Ge layer growth temperatures: lowering the growth temperature of the first layer and raising the growth temperature of the second layer in the epitaxial temperature range between 330 and 680 degree C led to further improvement in crystalline quality. Even further improvement was accomplished by high temperature annealing. Double crystal x-ray diffraction studies showed that the full width at half maximum (FWHM) of the (400) diffraction of the 500 nm thick Ge layer was 240 s. A Ge layer with a dislocation density on the order of 10**7 cm** minus **2 was obtained. A GaAs layer was subsequently grown by MOCVD. The FWHM of the 2 mu m thick GaAs layer was 140 s.