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We consider a system of R cubic forms in n variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided n ≥ 25 ⁢ R {n\geq 25R} , we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular, we obtain the Hasse principle for systems of cubic forms in 25 ⁢ R {25R} variables, previous work having required that n ≫ R 2 {n\gg R^{2}}. One conjectures that n ≥ 6 ⁢ R + 1 {n\geq 6R+1} should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport. [ABSTRACT FROM AUTHOR]