We report on the synthesis of nanostructured and crystalline tapered Si and Ge polyhedra via metal-catalyzed chemical vapor deposition. These Si and Ge nanocones (SiNCs, GeNCs) possess tips with near-atomic sharpness, micron-scaled bases, hexagonal cross-sections, and controllable apex angles. High-resolution transmission electron microscopy, selected-area electron diffraction and Raman scattering spectroscopy and analysis indicate that the SiNCs are of the diamond-hexagonal Si(IV) phase.