Lévy Matters IV : Estimation for Discretely Observed Lévy Processes Estimation and calibration of Lévy models via Fourier methods Adaptive Estimation for Lévy processes Parametric estimation of Lévy processes The aim of this volume is to provide an extensive account of the most recent advances in statistics for discretely observed Lévy processes. These days statistics for stochastic processes is a lively topic driven by the needs of various fields of application such as finance the biosciences and telecommunication. The three chapters of this volume are completely dedicated to the estimation of Lévy processes and are written by experts in the field. The first chapter by Denis Belomestny and Markus Reiß treats the low frequency situation and estimation methods are based on the empirical characteristic function. The second chapter by Fabienne Comte and Valery Genon Catalon is dedicated to non parametric estimation mainly covering the high frequency data case. A distinctive feature of this part is the construction of adaptive estimators based on deconvolution or projection or kernel methods. The last chapter by Hiroki Masuda considers the parametric situation. The chapters cover the main aspects of the estimation of discretely observed Lévy processes when the observation scheme is regular from an up to date viewpoint