Material Requirements Planning (MRP), a core component of enterprise resource planning (ERP) systems, is widely used by manufacturers to determine the production lot sizes of components. These lot sizes are typically computed based on deterministic and dynamic demand assumptions, while safety stocks, which hedge against demand uncertainty, are determined independently based on different assumptions. As the lot sizes and safety stocks are not determined simultaneously, sub-optimal decisions are often used in practice. The critical impact of inventories and service levels in manufacturing motivates the study of stochastic optimization methods for MRP. In this study, we investigate stochastic optimization methods for MRP systems under demand uncertainty. A two-stage and a multi-stage model are proposed to deal with the static-static and static-dynamic decision frameworks, respectively. We first derive structural properties of the two-stage and multi-stage models to provide insights on the differences between the plans created with these two models. As multi-stage stochastic programs are not convenient in real-world applications, several practical enhancements are proposed. First, to address scalability issues, we employ heuristics in combination with advanced sampling methods. Second, to allow real-time static-dynamic decisions, we derive a policy from the solution of the multi-stage model. Third, to deal with the dynamic-dynamic decision framework, we employ a rolling horizon implementation. The effectiveness and performance of stochastic optimization for MRP are validated by numerical experiments, which demonstrate that the stochastic optimization approaches have the potential to generate significant cost savings compared to traditional methods for production planning and safety stocks determination.