  Although it is not known which groups can appear as torsion groups of
elliptic curves over cubic number fields, it is known which groups can appear
for infinitely many non-isomorphic curves. We denote the set of these groups as
$S$. In this paper we deal with three problems concerning the torsion of
elliptic curves over cubic fields. First, we study the possible torsion groups
of elliptic curves that appear over the field with smallest absolute value of
its discriminant and having Galois group $S_3$ and over the field with smallest
absolute value of its discriminant and having Galois group $\Z/3\Z$. Secondly,
for all except two groups $G\in S$, we find the field $K$ with smallest
absolute value of its discriminant such that there exists an elliptic curve
over $K$ having $G$ as torsion. Finally, for every $G\in S$ and every cubic
field $K$ we determine whether there exists infinitely many non-isomorphic
elliptic curves with torsion $G$.

Number Theory;Algebraic Geometry