We prove that every pseudocompact
topological Abelian group $G$ admits a pseudocompact topological
group topo\-logy with a non-trivial convergent sequence.
Imposing some restrictions on the properties of $G$, stronger
properties are also obtained. If, for instance, $G$ is an Abelian
group with $m(\beta)\leq r_0(G)\leq|G|\leq 2^\beta$ (see the
Introduction below for unexplained terminology) for some uncountable
cardinal $\beta$, and $X$ is any topological space with $|X|\leq
r_0(G)$ and $w(X)\leq \beta$, then $G$ admits a pseudocompact
topological group topo\-logy that contains $X$ as a subspace.
If, on the other direction, $G$ is a torsion Abelian group that
admits a pseudocompact group topology, then, for every sequence
$(a_n)_{n \in \mathbb{N}}$ of $G$ there exists a pseudocompact group
topology on $G$ for which some subsequence of $(a_n)_{n \in
\mathbb{N}}$ converges.