In this thesis, a comprehensive analytical and computational study of linear and nonlinear optical response of nanostructured two-dimensional (2D) and bulk materials is presented. The new numerical methods developed in this thesis facilitate the efficient and accurate design of new artificial optical materials and novel nonlinear optical devices. Moreover, the presented results and conclusions can provide a deeper theoretical understanding of different resonant, nonlinear optical phenomena in photonic nanostructures. Two computational electromagnetic methods to calculate the interaction of light with linear and nonlinear diffraction gratings and more general periodic structures have been developed. An efficient formulation of the rigorous coupled-wave analysis (RCWA), a modal frequency domain method, for accurate near-field calculations and for complex oblique structures has been proposed. This method has been implemented into a powerful commercial software tool and applied to calculate diffraction in several nanophotonic devices highly relevant to practical applications. Beyond this commercial implementation, the RCWA has been extended to describe linear and nonlinear optical effects in nanostructured 2D materials, with second- and third-harmonic generation being the most important nonlinear processes. A key feature of this formulation is that it is independent of the height of the 2D material, and only requires knowledge of its linear and nonlinear optical properties. Using this method, plasmon resonances of nanostructured graphene have been investigated, and tuneable Fano resonances have been explored to increase the nonlinear efficiency of heterostructures containing transition metal dichalcogenide monolayers and nanopatterned slab waveguides. The second thrust of the thesis was devoted to extending the so-called generalised source method (GSM) to the area of nonlinear optics. In particular, its mathematical formulation has been extended to incorporate second- and third-order nonlinear optical effects, and the proposed nonlinear GSM has been used to design and optimise multi-resonant photonic devices made of nonlinear optical materials. In addition, this advanced computational method facilitated the understanding of strong nonlinear optical activity in plasmonic nanostructures, and explained the multipolar nonlinear optical response of certain nonlinear metasurfaces
