This thesis consists of three essays in stochastic inventory systems. The first essay is on the impact of input price variability and correlation on stochastic inventory systems. For a general class of such systems, we show that the expected cost function is concave in the input price. From this, it follows that higher input price variability in the sense of the convex order always leads to lower expected cost. We show that this is true under a wide range of assumptions for price evolution, including cases with i.i.d. prices and cases where prices are correlated and evolve according to an AR(1) process, a geometric Brownian motion, or a Markovian martingale. In addition, the result holds in cases where there is just a single period. We also examine the impact of price correlation over time and across inputs, and we find that expected cost is increasing in price correlation over time and decreasing in price correlation across components. We present results of a numerical study that provide insights on how various parameters influence the effects of price variability and correlation. The second essay is on the optimal control of inventory systems with stochastic and independent leadtimes. We show that a fixed base-stock policy is sub-optimal and can perform poorly. For the case of exponentially distributed leadtimes, we show that the optimal policy is state-dependent and specified in terms of an inventory-dependent threshold function. Moreover, we show that this threshold function is non-increasing in the inventory level and characterized by at most m parameters. That is, once the threshold function starts to decrease it continues to decrease with a rate that is at least one. Taking advantage of this structure, we develop an efficient algorithm for computing these parameters. In characterizing the structure of the optimal policy, we rely on an application of the Banach fixed point theorem. We compare the performance of the optimal policy to that of simpler heuristics. We also extend our analysis to systems with lost sales and systems with order cancellations. The third essay is on the optimal policies for inventory systems with concave ordering costs. By extending the Scarf (1959} model to systems with piecewise linear concave ordering costs, we characterize the structure of optimal policies for periodic review inventory systems with concave ordering costs and general demand distributions. We show that, except for a bounded region, the generalized (s,S) policy is optimal. We do so by (a) introducing a conditional monotonicity property for the optimal order-up-to levels and (b) applying the notion of c-convexity. We also provide conditions under which the generalized (s, S) policy is optimal for all regions of the state space.