As one of the most fundamental concepts in wave physics, resonance can give rise to a lot of interesting phenomena including low frequency band gaps. Because of its “divergent” nature, resonance also adds complexity into the modeling, and may even cause the failure of some widely adopted theories like quasi-static homogenization. In this talk, I will introduce my contributions in modeling classical wave systems with resonances by emphasizing on two major aspects: homogenization and linear dispersion relations. I will start with a brief review of my work on effective medium theories that can deal with resonances and show how those theories can guide our explorations of zero-index materials by adjusting resonances. Then, I will reveal the link between a zero-index material and linear dispersion relations at the Brillouin zone center of periodic structures, and go beyond the effective medium theory to introduce a general physical picture based on perturbation to describe the linear dispersion relations. The so-called Dirac-like cones and double-Dirac cones as well as their links to the classical analogues of topological insulators will be discussed with two examples.