The thesis explores size-dependent continuum mechanics in plate structures. Assuming a power-law distribution, the study includes functionally graded materials where the material properties are changed through the thickness. First, the nonlocal stress-driven theory with a bi-exponential kernel is introduced for thin plates based on the Kirchhoff assumption. Explicit analytical solutions are obtained in terms of hypergeometric and Meijer functions for Kirchhoff's nanoplate with different edge support configurations, while the Galerkin-based finite element method is implemented to compute both flexural deformation and natural frequencies. The analysis is extended to axisymmetric responses of moderately thick functionally graded nanoplates based on the Mindlin plate theory, capturing shear deformation effects. The stress-driven nonlocal elasticity framework with a bi-exponential kernel is used to study plates under uniform loading and different boundary conditions. Finite element solutions reveal the sensitivity of displacements and frequencies to characteristic length scales, material gradients, aspect ratios, and boundary configurations. The comparison between Mindlin and Kirchhoff models demonstrates that the Mindlin theory more effectively captures nonlocal effects, particularly for small characteristic parameters. Finally, a novel stress-driven formulation based on Kirchhoff plate assumptions is developed, by incorporating the Gaussian kernel to account for nonlocal size-dependent behavior in both radial and circumferential directions. Despite the previous formulation, which requires additional constraints on radial curvatures, the newly developed formulation is free from these extra constitutive boundary conditions. Analytical and finite element solutions highlight key size-dependent features, including reduced transverse displacements with increasing nonlocal parameters while maintaining unchanged moments.