Let T = R / Z be the written additively circle group and u = (un) be a sequence of integers. Many authors in various areas of Mathematics gave their attention to the following subgroups of T and their subsets t u( T ) = { x ∈ T | unx → 0 } . These subgroups are known with various names, here I refer to these subgroups as topologically u-torsion subgroups, because of their strong connection with torsion subgroups. Here, be- sides these subgroups in the circle group, I consider their nat- ural generalization for an arbitrary topological abelian group, with particular attention to the compact case: for a topologi- cal abelian group X and a sequence of characters v = (vn) the following subgroup s v(X) = { x ∈ X | vn(x) → 0 } is called characterized subgroup. Here I present some of my research results. In particular, I give a complete description of the subgroups t u( T ) where u is an arithmetic sequence, that is a strictly increasing sequence where un | un+1 for every n ∈ N. I give also some new results on the study of the Borel complexity of these subgroups, both in the compact case and in the circle group. Moreover, I present a structure theorem for the subgroups that admit a finer locally compact Polish group topology. The latter is a sufficient condi- tion for a subgroup to be characterized. Furthermore, I give a complete description of closed characterized subgroups in arbi- trary topological abelian groups and various useful reductions to the metrizable case. Presenting these results, I take the op- portunity to give an exhaustive description of the state of the art in this topic and to show some applications to other areas of Mathematics, with the aim of providing a useful handbook to an expert audience and a starting point for potential research purposes to non-expert users.