Let \(F({\mathbf x})=\sum_{1 \leq i \leq n}F_ix_i^3\) denote a diagonal integral cubic form in \(n\) variables, and let \(N(P)\) denote the number of integral zeros of \(F\) in a box \(| x_i| \leq P\). Then the circle method in its classical form yields an asymptotic formula for \(N(P)\) of the shape \(cP^{n-3}+o\bigl(P^{n-3}\bigr)\) when \(n \geq 9\). Assuming a Riemann hypothesis for Hasse-Weil \(L\)-functions, such a formula was established when \(n=8\) or \(7\) by \textit{C. Hooley} [Acta Math. 157, 49--97 (1986; Zbl 0614.10038)]. Unconditional results on these problems with weaker forms of the error terms follow using the techniques of \textit{R. C. Vaughan} from [J. Reine Angew. Math. 365, 122-170 (1986; Zbl 0574.10046) and J. Lond. Math. Soc., II. Ser. 39, 205--218 (1989; Zbl 0677.10034)]. The author uses a new form of the circle method developed by him in [\textit{D. R. Heath-Brown}, J. Reine Angew. Math. 481, 149--206 (1996; Zbl 0857.11049)] from a device of \textit{W. Duke}, \textit{J. Friedlander} and \textit{H. Iwaniec} in [Invent. Math. 112, 1--8 (1993; Zbl 0765.11038)]. This is used in conjunction with Hooley's ideas, so he also needs to adopt a Riemann hypothesis for Hasse-Weil \(L\)-functions. On this basis he shows \(N(P) = O(P^{3+\varepsilon})\) when \(n=6\). Let \(r_3(n)\) denote the number of representations of \(n\) as a sum of three integer cubes. A corollary of the author's result is the bound \(\sum_{n \leq X}r_3^2(n) \ll X^{1+\varepsilon}\), which has also been established (on the same basis) by \textit{C. Hooley} in [Sieve methods, exponential sums, and their applications in number theory, Lond. Math. Soc. Lect. Note Ser. 237, 175--185 (1997; Zbl 0930.11071)]. The author also gives a result when \(n=4\), where the situation is different in that rational lines on the surface \(F(x)=0\) may contribute \(cP^2\) points to the surface. He shows that there are \(O(P^{3/2+\varepsilon})\) points on the surface other than those lying on rational lines, which his method can separate off. Again, this result needs a Riemann hypothesis. Of this, the part relating to the existence of a functional equation has been proved in some important cases by \textit{A. Wiles} in the course of work on Fermat's Last Theorem [Ann. Math. (2) 141, 443--551 (1995; Zbl 0823.11029)].