For a general nonsingular cubic fourfold $X\subset \PP^5$ and $e\geq 5$ an odd integer, we show that the space $M_e$ parametrizing rational curves of degree $e$ on $X$ is non-uniruled. For $e \geq 6$ an even integer, we prove that the generic fiber dimension of the maximally rationally connected fibration of $M_e$ is at most one, i.e., passing through a very general point of $M_e$ there is at most one rational curve. For $e < 5$ the spaces $M_e$ are fairly well understood and we review what is known.
"Cubic fourfolds and spaces of rational curves." Illinois J. Math. 48 (2) 415 - 450, Summer 2004. https://doi.org/10.1215/ijm/1258138390