Abstract In today's data-intense and complex business environment, inventory managers face the massive challenge of simultaneously optimizing millions of products' inventories. Due to the huge numbers of items and frequent changes in demand, simple and efficient inventory policies are particularly desirable. In the first part of this talk, we study inventory systems with lost sales and lead times, which have traditionally been considered intractable due to the curse of dimensionality. Recently, Goldberg et al. (2012) laid the foundations for a new approach to solving these models, by proving that as the lead time grows large, a simple constant-order policy (proposed earlier by Reiman (2004)) is asymptotically optimal. This was quite surprising, as it is exactly this setting (i.e. large lead times) that was previously believed intractable. However, the bounds proven there are impractical. The authors note that numerical experiments of Zipkin (2008) suggest that the constant-order policy performs quite well even for small lead times, and pose closing this gap (thus making the results practical) as an open problem. In this work, we make significant progress towards resolving this open problem and closing this gap. In particular, we prove that the optimality gap of the same constant-order policy actually converges exponentially fast to zero. We also derive simple and explicit bounds for the optimality gap, which make the result and methodology practical for realistic lead time values. In the second part of this talk, we investigate dual-sourcing inventory systems, which are notoriously difficult to optimize due to the complex structure of optimal solution and curse of dimensionality. Recently, so-called Tailored Base-Surge (TBS) policies have been proposed and analyzed in Allon and Van Mieghem (2010) and Janakiraman, Seshadri and Sheopuri (2014). Under such a policy, a constant order is placed at the regular source in each period to meet a base level of demand and the order placed at the express source follows an order-up-to rule to manage demand surges. Such TBS policies are conjectured to be asymptotically optimal as the lead time difference grows. In this work, we prove this conjecture by showing that when the lead time of the express source equals zero, a simple TBS policy is indeed asymptotically optimal as the lead time of the regular source grows large. Since many companies are incorporating such a simple and natural sourcing strategy in practice, our results can potentially contribute to its widespread use. Both works are joint with David A. Goldberg.