For many engineering problems, e.g. nanophotonic devices, high-order accurate numerical methods are desirable because they are potentially orders of magnitude more efficient than their low-order counterparts. However, realizing the potential payoff of high-order methods in complex domains with multiple materials, particularly for problems involving wave propagation, has proven challenging. In this talk I discuss our recent work on high-order accurate methods for wave systems, with a focus on dispersive and nondispersive Maxwell's equations. Complex geometry is treated with overset grids, and interfaces between different materials are accurately and efficiently treated using compatibility coupling conditions. Compatibility coupling is a key ingredient to our approach, and when combined with high-order accurate modified equation time stepping yields a scheme with large CFL-one time step restriction. In the context of light propagation, the techniques are applied to generalized dispersive material models and arbitrary nonlinear multi-level systems, which can be used for many plasmonic applications such as for ab initio time domain modeling of nonlinear engineered materials for nanolasing applications, where nano-patterned plasmonic dispersive arrays are used to enhance otherwise weak nonlinearity in the active media. Carefully verification is performed for a number of two and three-dimensional problems.