The category O of (modules of) a complex semisimple Lie algebra was introduced by J. Bernstein, I. Gelfand and S. Gelfand in 1976. Since then it has found applications and motivated much progress in areas as diverse as knot theory, D-modules, the geometric Langlands program and combinatorics. I will explain how much of the structure of category O can be explained by viewing it as a `reasonably faithful module' for the Weyl group (or rather the associated Hecke algebra). I will situate myself on the algebraic side of this story, make this statement precise and present some of my own contributions to the subject. To keep things as elementary and concrete as possible I will mainly focus on sl2 and sl3. Time permitting, I will say something about the algebraic geometry underlying this story.
