This chapter reviewed key elements of the theoretical foundation and numerical implementation of finite-difference time-domain (FDTD) solutions of Maxwell's equations. The chapter included: • Introduction and background • Review of Maxwell's equations • The Yee algorithm • The nonuniform Yee grid • Alternative finite-difference grids • Theory of numerical dispersion • Algorithms for improved numerical dispersion properties • Theory of numerical stability • Alternating-direction implicit time-stepping algorithm for operation beyond the Courant limit • Perfectly matched layer (PML) absorbing boundary conditions, including Berenger's split-field PML, the stretched-coordinate PML formulation, and the uniaxial anisotropic PML (UPML). With literally hundreds of papers on FDTD methods and applications published each year, it is clear that FDTD is one of the most powerful and widely used numerical modeling approaches for electromagnetic wave interaction problems. With expanding developer and user communities within an increasing number of disciplines in science and engineering, FDTD technology is continually evolving in terms of its theoretical basis, numerical implementation, and technological applications. The latter now literally approach the proverbial spectral range from dc to light.