א1-free groups, abelian groups whose countable subgroups are free, play a central role in Shelah’s celebrated proof that the Whitehead problem is un- decidable (1974). Since then, set-theoretic constructions of א1-free groups with additional properties have become a staple, providing examples and counterexamples to a number of important questions. In this talk, we will trace the larger history of א1-free groups and discuss some of the more re- cent results on the set-theoretic aspects of א1-free groups: the absoluteness of א1-freeness and the use of forcing to add a basis to an א1-free group.