The main goals of the thesis are: Introducing the Hilbert's 16th problem and its weak form; explaining the relation between the weak Hilbert's 16th problem and the problem of finding the zeros of Abelian integrals; introducing several methods towards finding the number of zeros of Abelian integrals; finally finding the upper bound for the number of limit cycles of some families of polynomial systems in terms of the number of zeros of their Abelian integral.