New meshless numerical methods for the analysis of plate deformation process, which are based on the local weighted residual, i.e. the Local Petrov-Galerkin (MLPG) method, are presented in this PhD thesis. Integration of the weak form of equilibrium equations over a local subdomain is carried out. The choice of the weight function in the weighted residual is arbitrary. Plates are considered as three dimensional bodies, which enables the use of the three-dimensional constitutive law. Discretization is carried out by nodes on both the upper and lower surfaces. The weight functions are chosen as the product of the Heaviside step function in the middle plane and linear polynomials in the direction perpendicular to the middle plane. The nodal unknowns are approximated by functions defined in the middle plane and polynomials over the plate thickness. Interpolating Moving Least Squares (IMLS) and Bspline approximation functions are applied in the middle plane. The applied functions are interpolating, i.e., the shape functions have the Kronecker delta property. A mixed formulation, with displacement and stress components approximated separatley by the same functions, is proposed. Discretizing the equilibrium equations with the nodal stress components yields a system of equations with more nodal unknowns than equations, and it is therefore necessary to introduce additional equations. These equations are the constitutive relations which connect nodal stress components with nodal strain components and the kinematic relations which connect nodal strain components with nodal displacement components. These relations have been discretized by nodal displacement components. After inserting these additional relations into the first system of equations, a system with an equal number of unknowns and equations is derived. Imposition of the displacement boundary conditions is carried out directly, analogously to the finite element method because of the interpolating shape functions. The traction boundary conditions are satisfied in integral sense. The Poisson thickness locking has been eliminated by approximating the quadratical transversal displacement component. The transversal shear locking has been eliminated by approximating the stress components. Accuracy of the proposed methods has been validated in numerical examples by comparing the results with available analythical solutions.