In this talk, beginning with the classical comparison principles in ODEs, PDES and FDEs, the concept of monotone dynamical system is introduced. Then its basic properties and the generic convergence to equilibrium or periodic points are explained. Finally, the most recent results on skew-product monotone semiflows of the speaker will be presented, that is, the dynamics on every omega limit set of monotone and uniformly stable skew-product semiflows is conjugate to a minimal and distal dynamical system, this result is applied to study the asymptotic almost periodicity of solutions to almost periodic reaction-diffusion equations and differential systems with time delays.