Abstract. Let µ 1 , . . . , µ s be real numbers, with µ 1 irrational. We investigate sums of shifted cubes F (x 1 , . . . , x s ) = (x 1 − µ 1 ) 3 + . . . + (x s − µ s ) 3 . We show that if η is real, τ > 0 is sufficiently large, and s 9, then there exist integers x 1 > µ 1 , . . . , x s > µ s such that |F (x) − τ | < η. This is a real analogue to Waring's problem. We then prove a full density result of the same flavour for s 5. For s 11, we provide an asymptotic formula. If s 6 then F (Z s ) is dense on the reals. Given nine variables, we can generalise this to sums of univariate cubic polynomials.