Today, study of systems of difference equatins is very important and necessary because they have many applications in sciences such as economy, biology and natural modeling problems. a category of systems of difference equations is rational systems that consist of a lot systems. a great family of these systems are systems of rational difference equations in the plane. In general, a system of rational difference equations in the plane is systems of the form: with non-negative parameters ,,,,,,,,,,,, and with arbitrary nonnegative initial conditions x0 and y0 such that the denominators are always positive. Each of the 12 parameters of this system is either positive or zero. Hence, there exist 2401 special cases of systems with positive parameters included in this system. The systematic study on this subject have started in 2009 by Garry Ladas and his cooperators. This thesis is an extension and generalization of the works done by Ladas and Kulenovic. First, in the preface part, we present a history and some economic and biologic applications of the systems of rational difference equations in the plane. Then, in the next part we give some basic definitions, theorems and propositions on maps and difference equations that we need to study on dynamics of rational systems in the plane and definations of competetive and cooperative systems. In chapter three, we present a perfect 1. The systems with both solutions bounded, 2. The systems with one solution bounded and one unbounded, 3. The systems with both solutions unbounded.