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. In this thesis we take a proactive approach to generate robust and stable schedules for the environments with two sources of uncertainty: processing time variability and machine breakdowns. The information about the uncertainty is modeled using cumulative distribution functions and probability theory is utilized to derive inferences. We first focus on the single machine environment. We define 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. We identify special cases for which the measures can be optimized without much difficulty. We develop a dominance rule and two lower bounds for one of the robustness measures, which are employed in a branch-and-bound algorithm to solve the problem exactly. We also propose a beam-search heuristic to solve large problems for all five measures. We provide extensive discussion of our numerical results. Next, we study the problem of optimizing both robustness and stability simultaneously. We generate the set of all Pareto optimal points via -constraint method. We formulate the sub-problems required by the method and establish their computational complexity status. Two variants of the method that works with only a single type of sub-problem are also considered. A dominance rule and alternative ways to enforce the rule to strengthen one of these versions are discussed. The performance of the proposed technique is evaluated with an experimental study. An approach to limit the total number of generated points while keeping their spread uniform is also proposed. Finally, we consider the problem of generating stable schedules in a job shop environment with processing time variability and random machine breakdowns. The stability measure under consideration is the sum of the variances of the realized completion times. We show that the problem is not in the class NP. Hence, a surrogate stability measure is developed to manage the problem. This version of the problem is proven to be NP-hard even without machine breakdowns. Two branchand-bound algorithms are developed for this case. A beam-search and a tabu-search based two heuristic algorithms are developed to handle realistic size problems with machine breakdowns. The results of extensive computational experiments are also provided.