The fundamental natural frequency is calculated of a shallow shell, or dished plate, having a square simply supported boundary. A Rayleigh-Ritz procedure is used for the case of small amplitudes, based on a simplified double sine series for transverse displacement. The dependence of fundamental frequency on initial central rise is given, and it is found that the transition between an extensional and a partly inextensional mode occurs at a central rise of about eight times the shell thickness. The complex relationship between frequency and large amplitudes is determined approximately for amplitudes sufficient to cause “snap-through”. The amplitude-frequency relationship for an initially flat plate is also determined and found to agree well with that obtained by other authors.