This thesis is based on the works done in [2,9,23]. In recent years , rational difference equations have attracted the attention of many researchers for many reasons . On the one hand , they are examples of nonlinear equations whose dynamics present complete and rich dynamics . On the other hand , rational equations frequently appear in som e biological models , and , hence , their study is of interest due to their applications . A good example of both facts is Riccati difference equatio The Riccati equations is very well-known to have very rich dynamics . The first order Riccati equation has been studied thoroughly in [13,14,16,17]. In this thesis we study the global behaviours of all solutions of the second order rational difference equations. I t is associated with a third order linear difference equation . This association and other features are used to study the global behaviour . solution. There may exist points whose infinite iteration of the Riccati equation is not well-defined Therefore, the set of all such points are called forbidden set . In order to determine the global behaviour of all solutions of second order Riccati equation it is necessary to determine forbidden set . We show that with non-negative coefficients most initial points in IR 2 generate solutions of the second order equation that converge to a positive fixed point . However , there are "rare" initial values that generate periodic solutions of all possible periods as well as non-periodic oscillatory solutions . In this thesis we study the global behaviors of all solutions of some rational difference equations of orders two and three containing quadratic terms . We determine the forbidden sets of each equation explicitly and show that for initial values outside the forbidden sets , their solutions may converge to 0 , a positive fixed point , a periodic point of period 2, or the solution is unbounded . Rational system of first-order difference equations in the plane are also studied. The problem of stability and periodicity for such systems may be reduced to the analysis of a related matrix equation . This fact is used to characterize the forbidden set . Since some coefficients can take negative values, the forbidden set is not empty. We obtain the forbidden set for oll cases. Furthermore, as a general result we prove that the forbidden set is a Lebesgue measurable set. Its Lebesgue measure is always zero. Last conclution is based on the works done in [7], [18]. This shows that the autonomous rational equations have a good set , in the sense that these equations have a big set of solutions . Thus , studying problems related to the behaviour of solutions , such as stability or periodicity , will be worthwhile .