I will talk about my recent result joint with S. Shelah establishing that the Borel space of torsion-free Abelian groups with domain $\omega$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989. Time permitting, I will also talk about some recent results (also joint with S. Shelah) on the existence of uncountable Hopfian and co-Hopfian abelian groups and on anti-classification results for the countable co-Hopfian abelian and $2$-nilpotent groups. In particular, we will see that the countable co-Hopfian groups are complete co-analytic in the Borel space of $2$-nilpotent groups with domain $\omega$, this solves an open question of Thomas, who posed the question for the space of all groups with domain $\omega$.