Exact expressions exist for the natural frequencies and mode shapes of the axisymmetric free vibrations of a spherical shell or dome. However, they are not easily amenable to numerical calculation and there are no reference standards of natural frequencies and modes available for use in engineering. The need for such reference standards is increasing nowadays because of the wide applications of shell structures in large building roofs, containers and vessels, and of the serious concern about their structural resistance to such dynamic forces as earthquakes and waves. Furthermore, the exact solution is even less practical for analysis of response to dynamic loading. This paper provides, first, the natural frequencies and modes for a wide range of conical angle, based on the exact solution. Compatible with these results, an approximate solution is then proposed which is suitable for dynamic response analysis and which easily provides natural frequencies and modal shapes which are accurate enough for engineering purposes.