  Let $E$ be an elliptic curve defined over $\Q$, and let $G$ be the torsion
group $E(K)_{tors}$ for some cubic field $K$ which does not occur over $\Q$. In
this paper, we determine over which types of cubic number fields (cyclic cubic,
non-Galois totally real cubic, complex cubic or pure cubic) $G$ can occur, and
if so, whether it can occur infinitely often or not. Moreover, if it occurs, we
provide elliptic curves $E/\Q$ together with cubic fields $K$ so that $G=
E(K)_{tors}$.
