Ring resonators provide a means of filtering specific wavelengths from a waveguide, and optionally dropping the filtered wavelengths into a second waveguide. Both of these features are potentially useful for astronomical instruments. In this paper we focus on their use as notch filters to remove the signal from atmospheric OH emission lines from astronomical spectra. We derive the design requirements for ring resonators for OH suppression from theory and finite difference time domain simulations. We find that rings with small radii (<10 μm) are required to provide an adequate free spectral range, leading to high index contrast materials such as Si and Si3N4. Critically coupled rings with high self-coupling coefficients should provide the necessary Q factors, suppression depth, and throughput for efficient OH suppression, but will require post-inscription tuning of the coupling and the resonant wavelengths. The overall prospects for the use of ring resonators in astronomical instruments is promising, provided efficient fibre-chip coupling can be achieved.