We consider single machine scheduling problems with a non-renewable resource. This type of problems has not been intensively investigated in the literature so far. For several problems of this type with standard objective functions (namely the minimization of makespan, total tardiness, number of tardy jobs, total completion time and maximum lateness), we present some complexity results. Particular attention is given to the problem of minimizing total tardiness. In addition, for the so-called budget scheduling problem with minimizing the makespan, we present some properties of feasible schedules.