Аннотация:In this paper we develop non-linear ADER schemes for time-dependent scalar linear and non-linear conservationlaws in one-, two- and three-space dimensions. Numerical results of schemes of up to fifth order of accuracy in bothtime and space illustrate that the designed order of accuracy is achieved in all space dimensions for a fixed Courantnumber and essentially non-oscillatory results are obtained for solutions with discontinuities. We also present preliminary results for two-dimensional non-linear systems.