We demonstrate second- and third-harmonic generation in a centrosymmetric CMOS-compatible material using ring resonators and integrated optical waveguides. The χ(2) response is induced by using the nanoscale structure of the waveguide to break the bulk symmetry of silicon nitride (Si3N4) with the silicon dioxide (SiO2) cladding. Using a high-Q ring resonator cavity to enhance the efficiency of the process, we detect the second-harmonic output in the visible wavelength range with milliwatt input powers at telecom wavelengths. We also observe third-harmonic generation from the intrinsic χ(3) susceptibility of the silicon nitride. Phase matching of the harmonic processes occurs due to the near coincidence of indices of refraction of the fundamental mode at the pump frequency and the corresponding higher-order modes of the harmonic fields.