Counting rational points on hypersurfaces For any n ≧ 2, let F ∈ ℤ [ x 1, … , xn ] be a form of degree d≧ 2, which produces a geometrically irreducible hypersurface in ℙn–1. This paper is concerned with the number N(F;B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F; B) = O(B n− 2+ ε ), whenever either n ≦ 5 or the hypersurface is not a union of lines. Here the implied constant depends at most upon d, n and ε. 584 83 - 115 83 - 115 Walter de Gruyter and Co