Nonlinear degenerate diffusion appears in the models of population dynamics when population density pressure is taken into account. In this talk we present a variety of mathematical models of this type, characterized by different physical (biological/ecological) assumptions. In the case of Fisher-type equations, including the ones that incorporate the chemotaxis effect, we derive a rigorous asymptotics for the sharp moving fronts. For particular models including 2-species competition and for predator-prey systems we present new examples of the exact traveling front solutions compactly supported on one side.