In this talk I will discuss some of the mathematical theories on nonlinear parabolic equations motivated by the desire to better understand the propagation phe- nomena. The talk will start with a brief review of classical works of Fisher, Kolmogorov- Petrovsky-Piskunov and Aronson-Weinberger, and then focus on recent results on free boundary models with local as well as nonlocal diffusion, which are variations of the reaction-diffusion models in the classical works.