The purpose of this paper is to apply elementary methods of classical analysis in proving some general uniqueness theorems concerning finite axisymmetric deformations of thin shells of revolution. Problems where the radial membrane stress is nonnegative are investigated, and it is rigorously shown that the solutions of the relevant boundary value problems for both closed and open shallow shells acted upon by arbitrary normal surface load and subjected to various edge conditions are unique. It is shown that this result also holds for a restricted class of boundary value problems for nonshallow shells. Furthermore, new uniqueness results are obtained for ring shells, including annular plates as a special case. It is mentioned by the author that the von Kármán equations for annular plates have not been analyzed in previous work with regard to the mathematical questions of existence and uniqueness of solutions. For annular flat membrane problems, these questions have recently been treated by the author and others.