Monolithic integration of III-V on silicon has been a scientifically appealing concept for decades. Notable progress has recently been made in this research area, fueled by significant interests of the electronics industry in high-mobility channel transistors and the booming development of silicon photonics technology. In this review article, we outline the fundamental roadblocks for the epitaxial growth of highly mismatched III-V materials, including arsenides, phosphides, and antimonides, on (001) oriented silicon substrates. Advances in hetero-epitaxy and selective-area hetero-epitaxy from micro to nano length scales are discussed. Opportunities in emerging electronics and integrated photonics are also presented.