In this paper we present a detailed study of the theory of free, axisymmetric vibrations of thin elastic spherical shell and demonstrate by experiment that the normal modes of vibration predicted by theory do exist. The theory, which is an expansion of an ancient study by Lamb, predicts the existence of two infinite sets of normal modes, one of which is bounded in frequency and the other unbounded. The first four modes in each set are identified by experiments on a small steel shell.