Abstract: We begin by describing recent work on Clemens' conjecture, namely, a proof of the fact that on a general quintic threefold F in P^4, there are only finitely many smooth rational curves of degree 11, that each curve C is embedded in F with generically-splitting normal bundle, and that there are no singular rational curves. The proof is based on an analysis of generic initial ideals of curves in P^4, which we use to bound the fiber dimension of an incidence correspondence of curves on quintics. We will also describe an approach to proving naive dimension estimates for rational curves on general hypersurfaces based on tropical geometry. Tristram Bogart and I have already applied these methods with some success to the case of a general quintic surface in P^3 (which we know contains no rational curves).