Convergent sequences in topological groups

We survey recent developments concerning the role of convergent sequences in topological groups. We present the Invariant Ideal Axiom and announce its effect on convergence properties in topological groups, in particular, the consistency of the fact that every countable sequential topological group is either metrizable or kω. We also outline a construction of a countably compact topological group without non-trivial convergence sequences using iterated ultrapowers of Bohr topology on a Boolean group from a selective ultrafilter and announce a related ZFC construction of such a group. We recall the main open questions in the area and formulate several new ones.