  We study the existence of a Chow-theoretic decomposition of the diagonal of a
smooth cubic hypersurface, or equivalently, the universal triviality of its
${\rm CH}_0$-group. We prove that for odd dimensional cubic hypersurfaces or
for cubic fourfolds, this is equivalent to the existence of a cohomological
decomposition of the diagonal, and we translate geometrically this last
condition. For cubic threefolds $X$, this turns out to be equivalent to the
algebraicity of the minimal class $\theta^4/4!$ of the intermediate Jacobian
$J(X)$. In dimension $4$, we show that a special cubic fourfold with
discriminant not divisible by $4$ has universally trivial ${\rm CH}_0$ group.
