In practice, scheduling systems are subject to considerable uncertainty in highly dynamic operating environments. The ability to cope with uncertainty in the scheduling process is becoming an increasingly important issue. This paper takes a proactive scheduling approach to study scheduling problems with two sources of uncertainty: processing time variability and machine breakdowns. Two robustness (expected total flow time and expected total tardiness) and three stability (the sum of the squared and absolute differences of the job completion times and the sum of the variances of the realized completion times) measures are defined. Special cases for which the measures can be easily optimized are identified. A dominance rule and two lower bounds for one of the robustness measures are developed and subseqently used in a branch-and-bound algorithm to solve the problem exactly. A beam search heuristic is also proposed to solve large problems for all five measures. The computational results show that the beam search heuristic is capable of generating robust schedules with little average deviation from the optimal objective function value (obtained via the branch-and-bound algorithm) and it performs significantly better than a number of heuristics available in the literature for all five measures.