Using Mie theory we derive a number of general results concerning the resonances of spherical and cylindrical dielectric antennas. Specifically, we prove that the peak scattering cross-section of radiation-limited antennas depends only on the resonance frequency and thus is independent of refractive index and size, a result which is valid even when the resonator is atomic-scale. Furthermore, we derive scaling limits for the bandwidth of dielectric antennas and describe a cylindrical mode which is unique in its ability to support extremely large bandwidths even when the particle size is deeply subwavelength. Finally, we show that higher Q antennas may couple more efficiently to an external load, but the optimal absorption cross-section depends only on the resonance frequency.