In this work we study robust scheduling problems assuming that the processing times can take any value in the budgeted uncertainty set introduced by Bertsimas and Sim (2003, 2004). We consider problems on a single machine that minimize the (weighted and unweighted) sum of completion times and problems that minimize the makespan on parallel and unrelated machines. We provide approximation algorithms: constant factor, average non-constant factor, (fully or not) polynomial time approximation schemes. In addition, we prove that the robust version of minimizing the weighted completion time on a single machine is in the strong sense.