It is known that the general solutions for elastic shallow spherical tnin shells without surface load can be expressed by means of the Kelvin function in terms of φ, stress function, and w, displacement normal to the snell surface. In this paper some impotant points which are deficient in the existing theories are revised: physical meanigs of mathematical solutions are considered and the terms giving multi-valued displacements are excluded, membrane stress and rigid-body displacement terms in the antisymmetrical case (n=1 whenw=w_n cos nθ) which can not be derived from the mathematical solutions, are added. Formulas for dending and membrane stresses and displacements are presented as complete solutions, and particular solutions for various loads and temperature distributions are tabulated. As specific applications, shallow sphericall shells for roof denies and large tracking antennae are discussed. In these examples equations for deciding the unknown constants are derived from boundary conditions in the form of matrix. They can readily be used in the design of the similar structures. In paragraphs 1 and 2, the fundamental equations of bending problem of shallow spherical shells without surface load (equilibrium and compatibility equations) are established. From paragraph 3 to 6, the general solutions are derived by means of the Kelvin function (ber, bei, ker, kei), and the fundamental character of this function is discussed. Furtner, cosidering on the nature of the stresses and displacements resulted from the solution as well as on the rigid body displacements, complete form of the solutions is accompished. Particular solutions for various loads and temparature conditions are taculated in paragraph 7 and complete solutions for shells without surface load are formulated in paragraph 8, both for stresses and displacements. In and after paragraph 9 are the applications of the solutions to the actual structures. Paragraph 9 is the example of a shallow spherical shell stbjected to exially symmetrical edge moment, and ressults from the shallow shell theory are compared with those from Hetenyi's approximate theory for non-shallow spherical shells. In peragraph 10 and 11, the reinforced shallow spherical shell roof subjected to various loads and a temperature distribution under several kinds of supporting conditions is discussed. Paragraph 12 is the example of a large antenna, the outer edge free and the inner edge clamped, under the follewing lord conditions: a. own weight in two standing states, vertical and horizontal to the ground. b. various wind loads. c. axially symmetrical and antisymmetrical distributions of temperature increment.