The flag f-vector of a graded poset P enumerates chains in P by their rank sets. More generally, the flag f-vector of a colored simplicial complex enumerates its faces by their color sets. I will discuss a variety of results that aim to give a description of flag vectors in various cases. In particular, there are results concerning the multiplicative behavior of extreme inequalities and bounds on the relative rates of growth for various color sets, as well as recent results of Frohmader indicating that getting a complete characterization may indeed be difficult in the general case. Curiously, both the Fibonacci and the Catalan numbers make an appearance in this story.