Abstract : In this paper, we study nonparametric estimation of the Lévy density for Lévy processes, first without then with Brownian component. For this, we consider 2n (resp. 3n) discrete time observations with step $\Delta$. The asymptotic framework is: n tends to infinity, $\Delta=\Delta_n$ tends to zero while $n\Delta_n$ tends to infinity. We use a Fourier approach to construct an adaptive nonparametric estimator and to provide a bound for the global L2-risk. Estimators of the drift and of the variance of the Gaussian component are also studied. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.