Let μ1,…,μs be real numbers, with μ1 irrational. We investigate sums of shifted cubes F(x1,…,xs) = (x1−μ1)3 + ⋯ + (xs−μs)3. We show that if η is real, τ>0 is sufficiently large, and s⩾9, then there exist integers x1>μ1,…,xs>μs 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(Zs) is dense on the reals. Given nine variables, we can generalize this to sums of univariate cubic polynomials.