This thesis deals with the bifurcations of limit cycles by perturbing some Hamiltonian systems. In many applications the number and positions of limit cycles are important to understand the dynamical behavior of the system. Using the idea of Poincare map and associated displacement function built on a proper segment transversal to the period annulus of the unperturbed system, the problem is reduced to the problem of finding maximum numbers of isolated zeros of some special Abelian integrals, so-called first order Melnikov function. We use Chebeychev criterion, asymptotic expansion of Melnikov function near graphics and center and theory of bifurcation to study Hopf bifurcations, Poincare bifurcations and bifurcation of limit cycles from some graphics, like Heteroclinic loop and by these we consider four ltr"