This talk is a survey of our recent studies on advection-reaction-diffusion equations with singular fast/slow diffusion. When the diffusion is degenerate with singular point u=0, the equations possess the so-called sharp traveling waves which are regionally degenerate. On the other hand, when the diffusion is fast diffusion, the equations become singular at u=0. The main concerns of this talk are to investigate the sharp traveling waves and their stabilities for slow diffusion case, and the stability of singular viscous shock waves for the fast diffusion case. Singularities for both the fast/slow diffusion cases cause the essential difficulty for the study.