We study the Lp-properties of positive Rockland operators and define Sobolev spaces on general graded groups. This generalises the case of sub-Laplacians on stratified groups studied by G. Folland in [3]. We show that the defined Sobolev spaces are actually independent of the choice of a positive Rockland operator. Furthermore, we show that they are interpolation spaces and establish duality and Sobolev embedding theorems in this context.