This thesis is an extension (and generalization) of the work done by Ping Bi and Shigui Ruan , " Bifurcation in delay differential equations and applications to tumor and immune system interactions models " In this thesis , we consider a two-dimensional delay differential system with two delays . By studing equlibirium points the stability regions for some different parameters of these points can be obtained and by analyzing the distribution of eigenvalues , linear stability of the equilibria and existence of Hopf , Bautin , and Hopf-Hopf bifurcations are obtained in which the time delays are used as the bifurcaion parameter . In this study we assume that our system has one positive equlibirium although it could have multipe positive equlibirium . For the sake of simplicity these bifurcations are considered for equal delays . Then we compute the center manifold and normal form of these bifurcations , by using of normal form we can use the fixed point of poincar$\\acute{e}$ map to show existence of periodic cycle in each bifurcation . The stabilty of limit cycle corresponding to the stability of fixed point of poincare map . General formula for the direction , period , and stability of the bifurcated periodic solutions are given for codimension two bifurcations , including Hopf bifurcation , Bautin bifurcation and Hopf-Hopf bifurcation . It is worth mentioning that , in this thesis Bautin bifurcation is assumed as a one-codimention bifurcation and it is because of Bautin bifurcation is obtained by degenerating Hopf bifurcation . In Hopf-Hopf bifurcation first we should study the conditions that there is not any resonant , then we study Hopf-Hopf bifurcation for one case of four cases . It is shown that the positive equlibirium is asymptotically stable if all two delays are less than their corresponding critical values and limit cycle occure if any one of these delays passes thruogh its critical value . Our main results illustrate that the delay has an important effect on the dynamical behaviors of the system . As an application , we study the dynamical behaviors of a model describing the interaction between tumor and cells and effector cells of the immune system . The model is described by a system of two differential equations with two delays which describe proliferation delay ( ) of tumor cells and the growth delay ( ) of immune effector cells stimulated by tumor cells . Numerical examples and simulations are presented to illustrate the obtained results .