Hetero-epitaxy of compound semiconductors on largely lattice mismatched substrates is discussed. To reduce dislocation density, microchannel epitaxy that is a growth technique consisting of selective area epitaxy and epitaxial lateral over-growth has been proposed. It is shown that the key issue of this technique is how to increase the lateral growth rate compared with the vertical growth rate, which means how one can enhance the growth anisotropy. In the case of liquid phase epitaxy, one can get a large anisotropy by using a facet on the top and atomically rough surfaces on the sides. With very small interface supersaturation, wide and flat microchannel epitaxial layers have been successfully obtained for GaAs and InP on Si substrates. Wide dislocation free areas have been obtained outside of a dislocated region just above the microchannel. Microchannel epitaxy by molecular beam epitaxy is also described.