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Over the past decade, remarkable advances have been realized in chip-based nonlinear photonic devices for classical and quantum applications in the near- and mid-infrared regimes. However, few demonstrations have been realized in the visible and near-visible regimes, primarily due to the large normal material group-velocity dispersion (GVD) that makes it challenging to phase match third-order parametric processes. In this paper, we show that exploiting dispersion engineering of higher-order waveguide modes provides waveguide dispersion that allows for small or anomalous GVD in the visible and near-visible regimes and phase matching of four-wave mixing processes. We illustrate the power of this concept by demonstrating in silicon nitride microresonators a near-visible mode-locked Kerr frequency comb and a narrowband photon-pair source compatible with Rb transitions. These realizations extend applications of nonlinear photonics towards the visible and near-visible regimes for applications in time and frequency metrology, spectral calibration, quantum information, and biomedical applications.