The thesis focuses on the analysis of various extensions of the classical multi-period single-item stochastic inventory problem. Specifically, we investigate two particular approaches of modeling risk in the context of inventory management: risk-averse models and robust formulations. We analyze the classical newsvendor problem utilizing a coherent risk measure as the objective function. Properties of coherent risk measures allow us to offer a unifying treatment of risk averse and min-max type formulations. We show that the structure of the optimal policy of the risk-averse model is similar to that of the classical expected value problem for both single and multi-period cases. The result carries over even when there is a fixed ordering cost. We expand our analysis to robust formulations of multi-period inventory problems. We consider both independent and dependent uncertainty sets and prove the optimality of base-stock policies for the general problem formulation. We focus on budget of uncertainty approach and develop a heuristic that can also be employed for a class of parametric dependency structures. We compare our proposed heuristic against alternative solution techniques.