Growing single-crystal semiconductors directly on an amorphous substrate without epitaxy or wafer bonding has long been a significant fundamental challenge in materials science. Such technology is especially important for semiconductor devices that require cost-effective, high-throughput fabrication, including thin-film solar cells and transistors on glass substrates as well as large-scale active photonic circuits on Si using back-end-of-line CMOS technology. This work demonstrates a CMOS-compatible method of fabricating high-quality germanium single crystals on amorphous silicon at low temperatures of <450 degrees C. Grain orientation selection by geometric confinement of polycrystalline germanium films selectively grown on amorphous silicon by chemical vapor deposition is presented, where the confinement selects the fast-growing grains for extended growth and eventually leads to single crystalline material. Germanium crystals grown using this method exhibit (110) texture and twin-mediated growth. A model of confined growth is developed to predict the optimal confining channel dimensions for consistent, single-crystal growth. Germanium films grown from one-dimensional confinement exhibit a 200% grain size increase at 1 mu m film thickness compared to unconfined films, while 2D confinement growth achieved single crystal Ge. The area of single crystalline Ge on amorphous layers is only limited by the growth time. Significant enhancement in room temperature photoluminescence and reduction in residual carrier density have been achieved using confined growth, demonstrating excellent optoelectronic properties. This growth method is readily extensible to any materials system capable of selective non-epitaxial deposition, thus allowing for the fabrication of devices from high-quality single crystal material when only an amorphous substrate is available.