Abstract : Topological optimization methods become very attractive for industrial applications. It becomes possible to satisfy more challenging specifications of industrial products by allowing modifications of the topology of the initial design. The most relevant topological optimization methods are based on the computation of a level set function: the material density in the case of the topological optimization via the homogenization theory, the built-in level set function in the level set method, and the topological gradient provided by the topological asymptotic expansion, which is the concern of this thesis. In the latter case, at convergence, the positivity of the topological gradient inside the final domain provides a necessary and sufficient optimality condition. To present the basic idea, we consider a domain Omega and j(Omega) = J(u_Omega) a cost function to be minimized, where u_Omega is the solution to a given PDE problem defined in Omega. We can generally prove that the variation of the criterion j(Omega - B(x, epsilon)) - j(Omega) is given by the asymptotic expansion (with respect to epsilon): f(epsilon)g(x)+o(f(epsilon)), where f(epsilon)>0 and tends to zero with epsilon. To minimize the criterion j, we just have to create infinitely small holes at some points where the topological gradient $g$ is negative. Such asymptotic formulas have already been obtained for various problems. In this thesis, we deal with the following topics: the insertion of small inhomogeneities in the domain, the case of non-homogeneous differential operators, arbitrary shaped hole and a hole at the boundary of the domain. The obtained results are illustrated by numerical experiments such as the optimization of waveguides.