For any $n \geq 3$, let $F ∈ ℤ[X 0,⋯,Xn]$ be a form of degree $d\geq 5$ that defines a non-singular hypersurface $X ⊂ ℙ n. The main result in this paper is a proof of the fact that the number $N(F;B) of ℚ-rational points on $X$ which have height at most $B$ satisfies $N(F;B)=Od, ε,n(Bn-1+ε) for any $\varepsilon >0$. The implied constant in this estimate depends at most upon $d$, $\varepsilon$ and $n$. New estimates are also obtained for the number of representations of a positive integer a...