Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900 , together with the other 22 problems . The original problem was posed as the problem of the topology of algebraic curves and surfaces . Actually the problem consists of two similar problems in different branches of mathematics : 1- An investigation of the relative positions of the branches of real algebraic curves of degree n ( and similarly for algebraic surfaces ). 2- The determination of the upper bound for the number of limit cycles in polynomial vector fields of degree n and an investigation of their relative positions. Usually , the maximum of the number of limit cycles is denoted by H(n) , and is called the Hilbert number . Recall that a limit cycle is an isolated closed orbit . It is the ?-(forward) or ?- (backward) limit set of nearby orbits . In many application the number and position of limit cycles are important to understand the dynamical behavior of the system . This problem is still open even for the case n=2 . Limit cycle behavior is observed in many physical and biological systems . The problem of determining when a nonlinear dynamical system exhibits limit cycle has been of great interest for more than a century . Limit cycles cannot occur in linear systems , conservative systems and gradient systems . The limit cycles are caused by nonlinearities. In mathematics , more specifically in the study of dynamical systems and differential equations , a Lienard equation is a second order differential equation , named after the French physicist Alfred-Marie Lienard . It was found that many oscillatory circuits can be modeled by the Lienard equation . It can be interpreted mechanically as the equation of motion for a unit mass . Application of Lienard's equation can be found in many important examples including chemical reactions , growth of a single species , predator-prey systems and vibration analysis. In this thesis , we study the maximum number of limit cycles in Hopf bifurcation for two types of Lienard systems, smooth and nonsmooth, and obtain an upper bound of the number . In some cases the upper bound is the least , called Hopf cyclicity . In fact, we stablish an algebraic method to compute the Liapunov constants and Hopf cyclicity for a general smooth Lienard system and a general nonsmooth Lienard system on the plane . In nonsmooth dynamical systems , we study the center-focus problem . In the Oualitative Theory of planar differential equations , the problem of determining whether a critical point with pure imaginary eigenvalues is a center or a weak focus is known as the center-focus problem . The solution of the center-focus problem for a particular system involves the knowledge of the sign of the so-called return map, P( ?), in some neighbourhood of the origin. This sign can be studied by computing the terms of the series expansion of P( ?) which can be obtained recurrently and are generically called the Liapunov constants. Also, we obtain a sufficient and necessary condition which ensures the origin is a center . In the end , we present some new and interesting applications with example .