Fano varieties are covered by rational curves. However, not every Fano variety is birational to a family of rational curves. Indeed, Koll?r showed that there exist complex Fano hypersurfaces that are not birational to conic bundles. In this talk, we extend his results to higher genus fibrations. We show that for every genus g, there exist complex Fano hypersurfaces that are not birational to fibrations by genus g curves. For example, we prove that a very general n-dimensional complex hypersurface of degree at least ~5n/6 does not birationally admit an elliptic fibration structure. Our proof uses Koll?r's degeneration to positive characteristic and Tate's genus change formula. This is joint work with Nathan Chen, Benjamin Church, and David Stapleton.