Aluminum nitride (AlN) is an appealing nonlinear optical material for on-chip wavelength conversion. Here we report optical frequency comb generation from high-quality-factor AlN microring resonators integrated on silicon substrates. By engineering the waveguide structure to achieve near-zero dispersion at telecommunication wavelengths and optimizing the phase matching for four-wave mixing, frequency combs are generated with a single-wavelength continuous-wave pump laser. Further, the Kerr coefficient (n₂) of AlN is extracted from our experimental results.