maximal number of the limit cycles of a planner polynomial vector field of degree N: X(x,y)dx+Y(x,y)dy=0. This is a very hard problem and it is not solved even for n=2 after over 110 years of research. Therefore Arnold suggested a weaker Problem which is now known as "Weak Hilbert's 16th problem" or "tangentical Hilbert's 16th problem" which seems far more tractable. In this weaker problem you are asked to find isolated zeros of associated "Abelian integral." Essential in many bifurcations problems we have to study limit cycles occurring in a near-Hamiltonian system, that appear when we make a small perturbation in a Hamiltonian systems with Hamiltonian function H(x,y). Then, associated to a given perturbation of the system, there exist a so-called first-order Melnikov function which is denoted by M(h) where h shows the level of energy in the corresponding Hamiltonian system. An essential step to study the weak Hilbert's 16th problem is to study the number of isolated zeros of M(h). The study of the asymptotic expansion of Melnikov functio at critical values is an interesting problem which is closely related to the weak Hilbert's 16th problem and limit cycle bifurcation near the corresponding singular cycle for near-Hamiltonian system. Thus, it is clear that the asymptotic expansion of M(h) is an essential ingredient for the study of bifurcation problems of near Hamiltonian system. However, except for the expansions near an elementary center or a homoclinic loop through a hyperbolic saddle point, it seems that few works has been done on the asymptotic expansions of M(h) at such a critical value $h_0$ that the level curve H(x,y)= contains a non-elementary singular point such as nilpotent singular points. In this thesis, based on papers by Han and Zhang , we introduced a short history of main results about Hilbert's 16th problem. In Chapter two we consider fundamental definitions and theorems that are necessary in other chapters. In chapter three we introduced Hilbert's 16th problem and its weak form, Abelian integrals and Hamiltonian systems. In chapter four the first-order Melnikov function of a homoclinic loop through a nilpotent saddle for general planar near-Hamiltonian systems is considered. The asymptotic expansion of this Melnikov function and formulas for its first coefficients are given. In the last chapter, the number of limit cycles which appear near the homoclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. An example is presented as an application of the main results.