A cornerstone theorem of free groups is the Nielsen-Schreier theorem - that every subgroup of a free group is itself free. In this talk we'll explore this result via Nielsen's proof and then discuss the statement of the Andrew-Curtis conjecture. This is an open problem that is a natural extension of the tools developed by Nielsen to prove the Nielsen-Schreier theorem, yet a proof has eluded mathematicians for nearly 50 years (and continues to do so!) The talk should be accessible to everyone with an interest in algebra.