The analysis presented in this paper makes use of the theory of integral equations to obtain solutions for certain prescribed boundary value problems from the theory of thin elastic plates. The deflection functions obtained satisfy both the partial differential equation of Lagrange throughout the region defined by the plate and the given boundary conditions and therefore are exact solutions to boundary value problems;In Chapter II the deflection functions for rectangular plates with mixed boundary conditions have been obtained by superposing line loads of unknown intensity onto a pinned rectangular plate. The intensity of these line loads is determined so that the plate will satisfy the clamped boundary conditions along the unknown load lines. In order to obtain the desired line load distribution it is necessary to solve a system of linear integral equations. The deflection functions obtained are general in that they hold for any load distribution which can be represented by a double Fourier series over the region of the original plate satisfying the mixed boundary conditions;In Chapter III the deflection functions for clamped sector shaped plates are determined by superposing unknown line load distributions along diametral lines of a clamped circular plate. The intensity distribution functions for the unknown line loads are determined by solving systems of linear integral equations which are the result of satisfying the prescribed boundary conditions along the unknown load lines. The deflection functions obtained hold in general for axially symmetric loads;In Chapter IV the problem of a finite thin elastic plate resting on a yielding subgrade is considered. It is assumed that the loading, plate and subgrade, possess axial symmetry and that the plate maintains continuous contact with the subgrade. The solution to this problem requires the solution of an integro-differential equation for the deflection function. A general solution is obtained for various types of loadings and differing types of elastic subgrades.