Dynamical systems theory is a powerful tool to analyze and understand the dynamical evolutions of a diverse range of problems in natural sciences and engineering . Most of the problems that have been considered so far were systems defined by a smooth function of its argument and there is a well-developed theory to analyze such systems qualitatively and geometrically . This approach has been proved to be very helpful for understanding many nonlinear phenomena in mathematics , physics , engineering , biology , etc . However , many significant phenomena are not smooth which occur everywhere such as electrical circuits that have switches , problems with friction and models in the social and financial sciences that a continuous change can cause discrete responses \\cite{Barbashin, Bernardo ,Ito,Filippov,Kunze} . Such problems are characterized by piecewise-smooth functions . They have a very rich dynamics and underlying mathematical structure . By making careful mathematical assumption consistent with the underlying physical assumption of the problem , it is possible to extend naturally many of the concepts once assumed to be in the domain of smooth dynamical systems . One of the important topics in the \\begin{equation}\\label{bd1} \\dot{x}=X_{n}(x,y), \\qquad \\dot{y}=Y_{n}(x,y) ,\ag{1} \\end{equation} where $ X_n $ and $ Y_n $ are polynomials of degree $ n $ . To date, this problem remains unsolved even for the quadratic case, i.e. $ n = 2 $ . We note that a linear system may have periodic orbits but has no limit cycle. In \\cite{VArnold} , Arnold proposed a weaker version of the problem by restricting \\eqref{bd1} to the form \\begin{equation}\\label{bd2} \\frac{dx}{dt}=-{\\partial H(x,y)\\over \\partial y}+\\varepsilon f(x,y),\\qquad \\frac{dy}{dt}={\\partial H(x,y)\\over \\partial x}+\\varepsilon g(x,y) ,\ag{2} \\end{equation} where $ \\varepsilon $ is a small parameter, $ H $ is a polynomial of degree $ n + 1 $ and $ f $ and $ g $ are polynomials of degree less than or equal to $ n $ . Thus, \\eqref{bd2} is a perturbation of the Hamiltonian system \\begin{equation}\\label{bd3} \\frac{dx}{dt}=-{\\partial H(x,y)\\over \\partial y},\\qquad \\frac{dy}{dt}={\\partial H(x,y)\\over \\partial x} .\ag{3} \\end{equation} To tackle the Hilbert 16th problem for \\eqref{bd2}, besides the Hopf bifurcation and homoclinic (or heteroclinic) bifurcation analysis, a crucial step is to study the cyclicity of period annulus of $ \\X_H $ which is roughly the total number of limit cycles (counting multiplicity) that can be bifurcated from a period annulus (or annuli) of \\eqref{bd3}. On the other side, the study of the number of limit cycles for discontinuous differential systems in the plane can be seen as an extension of the Hilbert 16th problem . But the n aive ideas of extending Hilbert’s 16th problem to non-smooth systems should be discarded since it was shown in $ 2015 $ that infinitely limit cycles are possible even in a piecewise linear system \\cite{Llibre1} . Because of the importance of this problem, in this thesis, we investigate the limit cycles and monotonosity of Abelian integrals in some smooth and non-smooth systems that plays an important role in the dynamic analysis of such system s .