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.