In this paper we discussed properties of monotone dynamical systems with a special focus on the Frenkel-Kontorova Model. We modeled dynamics of a model by the system of ordinary differential equations. A monotone dynamical system is a dynamical system on a metric space which has a property that ordered initial states lead to ordered subsequent states. We say that a system of ordinary differential equations generates the semiflow. We studied convergence criteria of the semiflow and some properties of the omega limit set. Furthermore, we defined competitive and cooperative differential equations. We say that the system of ordinary differential equations is cooperative if it generates a monotone semiflow in the forward time direction and competitive if it generates monotone semiflow in the backward time direction. In the last chapter we focused on the main topic of this master thesis, the Frenkel-Kontorova Model. That is a model which describes the dynamics of a chain of particles interacting with the nearest neighbors in the presence of an external periodic potential. In addition, we applied an external force to the system. We concluded that when F = 0, there always exists an equilibrium for this model. On the other hand, we also proved that we can find a force F large enough so that there are no equilibria. Practical applications of Frenkel-Kontorova model are numerous and we described two of them in this thesis.