The method of solution presented in this paper makes use of solutions of the governing differential equations of linear three-dimensional elasticity theory. Boundary conditions are satisfied by collocation at points on the boundary. The procedure is developed and investigated to determine its applicability to the solution of the problem of free vibrations of isotropic, linearly elastic bodies of revolution of arbitrary shape and arbitrary boundary conditions. The problem is formulated in terms of a series of solutions of the governing differential equations in cylindrical co-ordinates. Results are obtained for a number of bodies of revolution. The results are compared with analytical and finite element solutions. The comparisons clearly show that the boundary collocation method gives accurate answers with a minimum of computational effort. The method gives excellent results for both axisymmetric and non-axisymmetric vibrations of hollow as well as solid bodies of revolution. Furthermore, by considering thin circular plates and thin cylindrical shells, it is shown that even for these extreme cases accurate answers are produced.