Meshless computational methods for the analysis of plate and shell structures are proposed in this thesis. The developed algorithms are based on the local PetrovGalerkin approach. A shell is considered as a three dimensional (3-D) solid continuum, and the solid-shell concept, which allows the implementation of complete 3-D material models, is employed. Geometry of the shell is described by employing a mapping technique, whereby the shell middle surface is defined mathematically exactly. Discretization is carried out by the couples of nodes located on the upper and lower surfaces of the structure. The governing equations are the local weak forms (LWF) of the 3-D equilibrium equations, which are written over the local sub-domains surrounding the node couples. The approximation of all unknown field variables is carried out by using the Moving Least Squares (MLS) approximation scheme in the inplane directions, while simple polynomials are applied in the thickness direction. Both the purely displacement-based (primal) and mixed formulations are proposed and special attention is given to the elimination of locking effects. Two different primal formulations are presented where only the displacement field is approximated. In both cases, the Poisson’s thickness locking effect is circumvented by adopting the hierarchical quadratic interpolation for the transversal displacement component. The transversal shear locking phenomenon is alleviated by applying a sufficiently high order of the in-plane MLS functions. In the mixed approach, appropriate strain and stress components are approximated separately from the displacement field. The nodal strain and stress values are then expressed in terms of the approximated displacements, and a global system of equations containing only the unknown nodal displacement variables is obtained. In the formulation for plates, thickness locking is eliminated by modifying the nodal values of the normal transversal strain component, while the transversal normal stress is approximated instead of the transversal normal strain in the algorithm for curved shells. In the thin structural limit, transversal shear locking is efficiently suppressed by means of the separate strains approximation. It is theoretically proved that the mixed approach is numerically more efficient than the proposed primal meshless formulations. The numerical efficiency of the derived algorithms is demonstrated by numerical examples.