  Let $X \subset \mathbf{P}_{\mathbf{Q}}^{n-1}$ be a cubic hypersurface cut out
by the vanishing of a non-degenerate rational cubic form in $n$ variables. Let
$N(X,B)$ denote the number of rational points on $X$ of height at most $B$. In
this article we obtain lower bounds for $N(X,B)$ for cubic hypersufaces,
provided only that $n$ is large enough. In particular, we show that $N(X,B) \gg
B^{n-9}$ if $n \geq 39$, thereby proving a conjecture of T. D. Wooley for
non-conical cubic hypersurfaces with large enough dimension.
