  Assuming the existence of $\mathfrak c$ incomparable selective ultrafilters,
we classify the non-torsion Abelian groups of cardinality $\mathfrak c$ that
admit a countably compact group topology. We show that for each $\kappa \in
[\mathfrak c, 2^\mathfrak c]$ each of these groups has a countably compact
group topology of weight $\kappa$ without non-trivial convergent sequences and
another that has convergent sequences.
  Assuming the existence of $2^\mathfrak c$ selective ultrafilters, there are
at least $2^\mathfrak c$ non homeomorphic such topologies in each case and we
also show that every Abelian group of cardinality at most $2^\mathfrak c$ is
algebraically countably compact. We also show that it is consistent that every
Abelian group of cardinality $\mathfrak c$ that admits a countably compact
group topology admits a countably compact group topology without non-trivial
convergent sequences whose weight has countable cofinality.
