Let X be a general hypersurface over the field of complex numbers, and let R_e(X) be the space of smooth rational curves of degree e on X. When the degree of the hypersurface is very low compared to its dimension, it is known that R_e(X) is rationally connected. What can be said about the Kodaira dimension of R_e(X) when the degree of X is not very small? I will show that these spaces are not uniruled, and I will discuss some other results in this direction.