This thesis is divided in two main parts. In the first part which is based on an article by Maoan Han, Hong Zang and Junmin Yang (Limit Cycle Bifurcations by Perturbing a Cuspidal Loop in a Hamiltonian System, 2009), we study the analytical property of the first Melnikov function for general Hamiltonian Systems exhibiting a cuspidal loop of order (m) its asymptotic expansion at the Hamiltonian value corresponding to the loop. Then by using the first coefficients of this expansion we give some conditions for the perturbed system to have 4, 5 or 6 limit cycles in a neighborhood of the loop. As an application of our main results, we consider some polynomial Lienard systems of order three and find 4, 5 or 6 limit cycles. In the second part which is based on an article by Hongyan Ma and Maoan Han(Limit Cycles of a Z3-equivariant near-Hamiltonian System, 2009), we have studied the number of a limit cycles of a Z3-equivariant near Hamiltonian system of degree 3 and 4 which are a perturbation of a cubic Hamiltonian system. By using the Melnikov method, we respectively get 4 and 10 limit cycles.