Liénard systems are very important mathematical models describing oscillatory processes arising in applied sciences . In this thesis , we study polynomial Liénard systems of arbitrary degree on the plane , and develop a new method to obtain a lower bound of the maximal number of limit cycles based on paper . Using the method and basing on some known results for lower degrees , we obtain a new estimation for the number of limit cycles in these systems which greatly improve existing results . Consider a polynomial Liénard system of the form \\begin{equation*} \\dot{x}=y,\\,\\,\\,\\dot{y}=-g(x)-\\varepsilon f(x) y , \\end{equation*} where \\varepsilon is a small parameter , f(x) and g(x) are polynomials in x of degree n and m , respectively . The above system is called a Liénard system . It describes the dynamics of systems of one degree of freedom under existence of a linear restoring force and a nonlinear damping . It was shown by Liénard that under some conditions on the functions f(x) and g(x) in the system arise auto-oscillations . In the first half of the last century models based on the Liénard system were important for the development of radio and vacuum tube technology . Nowadays the system is widely used to describe oscillatory processes arising in various studies of mathematical models of physical , biological , chemical , epidemiological,physiological , economical and many other phenomena . Our study is devoted to finding Liénard systems which admit not a single , but few auto-oscillatory regimes (limit cycles) . Let H(n,m) denote the maximal number of limit cycles on system on the plane for \\varepsilon sufficiently small . The lower bound of H(n,m) for the Liénard system has been widely studied . The first chapter is reface . We describe differential equation and results that scientist obtained . We describe limit cycles and Hilbert's 16th problems . The second part of Hilbert's 16th problem concerns the uniform lower bounds on number of limit cycle that a polynomial system in plane can have . At the end of this chapter , we express some results about H(n,m) that had been already obtained . The second chapter devoted to present some basic definitions on theorems that we need to use in this thesis . In the third chapter , we study the number of limit cycles of polynomial Liénard systems . At first , we describe property Z(n,m,k) and prove some lemmas and theorems . After that , by helping these lemmas and theorems , we will prove some estimates of H(n,m) for fixed m, some estimates of H(m,m) and some estimates of H(m\\pm r,m) . In the forth chapter , we study limit cycles of some polynomial Liénard systems . Using the Melnikov functions about homoclinic loop and Heteroclinc cycles and bifurcation theory , we prove that H(2,5)\\geq 3,H(4,5)\\geq 5,H(6,5)\\geq 10,H(8,5)\\geq 10 . Next we consider a given Liénard system with g(x)=x(x^2-1)(x^2-\\alpha^2) with 0 \\alpha \\frac{\\sqrt{3}}{3} that will ensure that local maximum of the Hamiltonian function of with _{\\varepsilon=0} at the value x=1 will be positive and two heteroclinic orbits exist for \\varepsilon=0 . Without less of generality , we will take \\alpha=\\frac{1}{2} then the system becomes \\[\\dot{x}=y,\\,\\,\\,\\dot{y}=-\\frac{1}{4}x-\\frac{5}{4}x^3+x^5-\\varepsilon f(x,\\delta) y,\\] where \\[f(x,\\delta)=a_0+a_1 x^2+a_2 x^4+a_3 x^6+a_4 x^8\\] is a polynomial of degree 8 , \\delta=(a_0,a_1,a_2,a_3,a_4)\\in \\mathbb{R}^5 . Under some further assumptions by proving some lemmas and theorems , we give a lower bound for the number of H(8,5). In appendix , we provide the Maple codes for some of our computations .