In this paper we first develop an approximate theory for the leading order concentration dependence of the sedimentation coefficient for rodlike colloids/polymers/macromolecules. To first order in volume fraction phi of rods, the sedimentation coefficient is written as 1+alpha phi. For large aspect ratios L/D (L is the rod length, D its thickness) alpha is found to vary like proportional to>(*) over bar * (L/D)(2)/ln(L/D). This theoretical prediction is compared to experimental results. Then, experiments on fd virus are described, both in the isotropic and nematic phase. First-order in concentration results for this very long and thin (semiflexible) rod are in agreement with the above-mentioned theoretical prediction. Sedimentation profiles for the nematic phase show two sedimentation fronts. This result indicates that the nematic phase becomes unstable with the respect to isotropic phase during sedimentation.