Recent developments in chip-based nonlinear photonics offer the tantalizing prospect of realizing many applications that can use optical frequency comb devices that have form factors smaller than 1 cm(3) and that require less than 1 W of power. A key feature that enables such technology is the tight confinement of light due to the high refractive index contrast between the core and the cladding. This simultaneously produces high optical nonlinearities and allows for dispersion engineering to realize and phase match parametric nonlinear processes with laser-pointer powers across large spectral bandwidths. In this Review, we summarize the developments, applications and underlying physics of optical frequency comb generation in photonic-chip wave-guides via supercontinuum generation and in microresonators via Kerr-comb generation that enable comb technology from the near-ultraviolet to the mid-infrared regime.