For any integers $d,n \geq 2$, let $X \subset \mathbb{P}^{n}$ be a non-singular hypersurface of degree $d$ that is defined over the rational numbers. The main result in this paper is a proof that the number of rational points on $X$ which have height at most $B$ is $O(B^{n - 1 + \varepsilon})$, for any $\varepsilon > 0$. The implied constant in this estimate depends at most upon $d$, $\varepsilon$ and $n$.