Diffractive optics is an emerging technology with many practical applications. The significant applications have driven the need for mathematical models and numerical algorithms: to provide rigorous and accurate solutions of the full electromagnetic vector field equations for complicated grating structures, thus predicting performance given the structure, and to carry out optimal design of new structures. The former situation is often referred to as the direct problem, while the latter case is an example of the inverse problem. This talk is concerned with recent mathematical developments on direct and inverse problems in the modeling of diffractive optics. For the direct problems, issues on model formulation, well-posedness of the models, and convergence analysis will be addressed. Concerning the inverse problems, the speaker will present recent results on uniqueness, stability, and reconstruction. Related ongoing research projects in diffractive optics and electromagnetics will also be discussed.