The epitaxial growth of GaAs on Si substrates through the use of a Ge/graded Si1-xGex/Si buffer layer would allow monolithic integration of GaAs-based opto-electronics with Si microelectronics. As an initial step toward this goal, this study examines factors which influence the quality of GaAs growth by molecular beam epitaxy (MBE) on bulk Ge substrates. Key findings include the need for an epitaxial Ge smoothing cap deposited in the MBE chamber, the significant detrimental effect of As overpressure on the resultant GaAs crystalline quality, and the efficiency of a very thin (similar to 3 nm) migration enhanced epitaxy (MEE) nucleation layer at suppressing both anti-phase domain (APD) formation and interdiffusion across the GaAs/Ge heterointerface. Using this developed optimized growth process, APD-free GaAs on Ge is obtained which has undetectable Ga and Fe cross-diffusion, and As diffusion into the substrate at less than or equal to 1 x 10(18) cm(-3). Preliminary results for growths on Ge/Si1-xGex/Si substrates are also presented.