Assortment planning is a major operational issue that arises in many industries, such as retailing, airlines and consumer electronics. Given a set of products (or services) that are differentiated by price, quality and possibly other attributes, one has to decide on the subset of products and the respective quantities that will be stocked and offered to customers who usually exhibit substitution behavior. We study several assortment models with and without dynamic substitution and show how different assumptions on the underlying customer choice model translate to the complexity of the model. We also discuss several important cases that admit efficient optimal algorithms, as well as cases where there exists polynomial time approximation scheme (PTAS). Interestingly, some of the near optimal solutions are based on sparse assortments. The talk is based on several papers with Ali Aoud, Vivek Farias, Vineet Goyal and Danny Segev.