This thesis can be divided into two main parts. In the first part, we start with a brief introduction to discrete-time dynamical systems and bifurcation theory. Invariant sets such as fixed points and periodic orbits are defined and the notion of stability of invariant sets is introduced. By defining the topological equivalence of dynamical systems, we are equipped with the required settings to introduce the notions of bifurcation and bifurcation diagram. Then the numerical methods for the local bifurcation analysis of discrete-time dynamical systems are discussed. Using Newton like methods, the fixed points of maps are located. By releasing the first parameter these fixed point are continued to attain a curve of fixed points. While continuing these fixed points we search for singularities such as bifurcation and branching points. When a bifurcation point is located a second parameter is released with the aim of continuing and finding co-dimension 2 bifurcations. In the last section of this part, by using MATCONTM toolbox i MATLAB some practical examples are studied and their bifurcations are located, found and continued. The delayed logistic map is analyzed using MATCONT toolbox. Phenomena such as Phase Locking are observed. Neimark-Sacker, Fold and Flip bifurcation curves are computed and plotted. The bifurcation diagram for the delayed logistic map is also drawn. The second part, having a similar structure, starts with the computation of the stable and unstable invariant manifolds and finding their intersections, aiming at locating the connecting orbits. Two algorithms for the computation of Global Invariant Manifolds are introduced and implemented in Matlab. The algorithm for finding the intersections is capable of finding the connecting orbits of maps such as heteroclinic and homoclinic connections. By employing a method called the continuation of invariant suaces, we introduce and implement an algorithm for the continuation of connecting orbits and the computation of Fold bifurcation curve corresponding to the tangency of invariant manifolds. Algorithms for the continuation of connecting orbits are implemented in Matlab. The Jacobian Matrix of the continuation problem defining system is computed symbolically and in details. In the end, once again MATCONTM is utilized to continue the connecting orbits of some well-known maps. In order to find the Limit Point bifurcation of connecting orbits, the second iteration of the Generalized Henon map heteroclinic connections are computed and continued. The Limit Point Bifurcation curve is also computed. Homoclinic connections in Maps such as Generalized Henon, Shear and Ikeda are computed and continued. To find these homoclinic connections we needed to compute the stable and unstable manifolds and find their intersection points.