This thesis deals with the number of limit cycles for small quadratic perturbations of quadratic integrable systems: Where ? is a small parameter and f(x,y,?), g(x,y,?) are quadratic polynomials in x,y with coefficients depending analytically on ?, H(x,y) is the first integral of the unperturbed system with the integrating factor M(x,y). The plane quadratic systems with at least one center are always integrable, which can be 14.4pt; HEIGHT: 15.6pt" id=_x0000_i1025 type="#_x0000_t75" , reversible , codimension four and generalized Lotka–Volterra . A natural problem is asking for a maximal number of limit cycles, bifurcated from the period annulus of these four 14.4pt; HEIGHT: 15.6pt" id=_x0000_i1025 type="#_x0000_t75" , this number is closely related to the weak Hilbert 16th problem in the quadratic case. It is well known that the weak Hilbert 16th problem for Hamiltonian , was solved completely by many authors, see [4, 13] and the references therein. The next natural step is to find the upper bound of the number of limit cycles which bifurcated from the period annulus of quadratic integrable but non-Hamiltonian systems, under quadratic perturbation. As usual, we use the notion of cyclicity for the total number of limit cycles which can emerge from a configuration of trajectories (center, period annulus, a singular loop) under a perturbation. In [22, 58], the authors proposed a conjecture about the cyclicity of the period annulus of quadratic centers , and under small quadratic perturbation.