We develop a unifying framework to prove the existence of optimal policies for a large class of inventory systems. The framework is based on the transformation of the inventory control problem into a game, each round of which corresponds to a single replenishment cycle. By using parametrized optimization methods we show that finding the equilibrium in this game is equivalent to finding the optimal long-run average cost of the inventory problem, and demonstrate that any root-finding algorithm can be used to find the optimal inventory policy. We then cast the associated parametrized problem into an optimal stopping problem. The analysis of this problem reveals that the structure of the optimal policy is a geometrically obvious consequence of the shape of the inventory cost function. It also allows us to derive bounds on the optimal policy parameters and devise a straightforward scheme to efficiently compute them. The proposed framework enjoys the power tools of optimal stopping theory and thereby provides a new and powerful line of attack that can address periodic and continuous review inventory problems with discrete and continuous demands in a unified fashion.