In this talk I will discuss some of the mathematical theories on parabolic equations motivated by the desire of providing better models for various propagation phenomena. I will start by recalling briefly classical works of Fisher, Kolmogorov-Petrovski-Piskunov and Skellam; then I will present some recent results on models with free boundaries and compare them with classical results of Aronson-Weinberger and others; finally I will discuss the extensions of the theories (initially established in homogeneous media) to heterogeneous media.