Kashiwara, Kazhdan, and Lusztig have introduced important bases of the Hecke algebra (the "Kazhdan-Lusztig" basis) and of the quantum coordinate ring of SL_n (the "crystal" or "dual canonical" basis). While these bases have very nice properties and have applications in representation theory, combinatorics, and algebraic geometry, no elementary description is known for either basis. We will discuss a factorization theorem for the Kazhdan-Lusztig basis and an improved description of the dual canonical basis which have led to recent results in the above fields.