This thesis studies inventory control with risk of major supply chain disruptions, specifically border closures and congestion. We first investigate an inventory system in which the probability distributions of order leadtimes are dependent on the state of an exogenous Markov process; we will model border disruptions via this exogenous process. We consider stationary, state-dependent basestock policies, which are known to be optimal for the system under study, and develop an expression for the long-run average cost of an arbitrary policy of this form. Restricting our attention to state-invariant basestock policies, we show how to calculate the optimal basestock (or order-up-to) level and long-run average cost. We provide a sufficient condition for the optimality of a state-invariant basestock policy and monotonicity results for the optimal state-invariant order-up-to level. We finally give the optimal state-invariant order-up-to level for a special class of supply states. Motivated by the possibility of port of entry closures in the event of a security incident, we specialize the previous model to a two-stage international supply chain. A domestic manufacturer orders a single product from a foreign supplier and the orders must cross an international border that is subject to closure. We first assume that border congestion is negligible. The manufacturer's optimal inventory policy and long-run average cost are analyzed. We present structural policy results and the results of a comprehensive numerical study that have important implications for business and for the cooperation between business and government in disruption management and contingency planning. Finally we extend the border closure model to include both border closures and the resulting congestion. We model the border processing system as a discrete-time, single-server queue with constant arrival rate and Markov-modulated service rate. A key task is the development of the leadtime distribution, which is more complex than in the previous model. We present the results of a comprehensive numerical study and provide managerial insights.