The research work undertaken in this dissertation is comprised of two parts. The first part is concerned about the comparative study of weak and strong meshless formulations for the numerical solution of elliptic boundary value problems. Meshless methods based on weak and strong formulation provide alternate numerical solution procedures to the well established solution techniques, such as finite element methods, finite difference methods, boundary element methods and finite volume methods. The meshless weak formulation considered in this dissertation is the well-known element free Galerkin method whereas, a strong meshless formulation chosen, is the local radial basis functions collocation method. The weak meshless formulation in the form of the element free Galerkin method is proposed while incorporating the new numerical quadrature technique based on multi-resolution Haar wavelets for approximating the displacement and strain modeled by elliptic boundary value problems in one- and two-dimensional spaces. The element free Galerkin method with numerical integration based on Gaussian quadrature has also been implemented for the numerical solution of the elliptic boundary value problems. In addition, a meshless collocation method has also been proposed in strong form using the local radial basis functions collocation method for the numerical solution of the elliptic boundary value problems. A comparative study in terms of accuracy and stability of both the weak and strong meshless formulations is carried out for the elliptic boundary value problems in one- and two-dimensional spaces. The second part of the research work undertaken in this thesis is focused on the applications of meshless methods for the solutions of solid mechanics and structural optimization problems. The element free Galerkin method with numerical integration based on Haar wavelets is also used to solve one- and two-dimensional elasto-static problems. Numerical solution obtained with these methods is in excellent agreement with analytical solution. Further, the element free Galerkin method is implemented with level set method for the two-dimensional structural optimization problems for minimum compliance design. The shape and topological sensitivities are obtained by the element free Galerkin method. The proposed topology optimization method is capable of automatically inserting holes during the optimisation process using the topological derivative approach. The structural geometry is updated through the numerical solution of modified Hamilton-Jacobi type partial differential equation by the level set method. Furthermore, the element free Galerkin method has also been combined with radial basis functions in the frame work of level set method for the solution of two-dimensional structural optimization problems. Furthermore, the finite element method has also been coupled with local radial basis functions based level set method. The original Hamilton-Jacobi equation has been transformed into a system of coupled ordinary differential equations. To highlight the associated advantages and disadvantages of the radial basis functions in the framework of level set method, a comparative study has also been carried out. The proposed methods are implemented for the optimal solutions of different types of structures with application of single and multiple loads.