This thesis deals with the bifurcations of limit cycles by perturbing some Hamiltonian systems. To study the perturbations of Hamiltonian systems, the first order Melnikov function plays an important role. By finding its zeros, we can find limit cycles. Using he asymptotic expansions of the Melnikov function near a Hamiltonian value corresponding to an elementary center, a nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points we study the bifurcation problem. To study the Poincar\\'{e} bifurcation we may use the Chebysheve criterion developed by Villadelprat.