The idea of using geometric objects to represent subgroups of the free
group goes back to the last quarter of the twentieth century with the
work of Serre and especially Stallings. It soon became clear that the
original (topological) approach admitted a neat and appealing
reformulation in terms of automata (now called Stallings automata).
This theory has become the main modern tool for solving algorithmic
questions on subgroups of the free group and has provided insightful
proofs to innumerable results, both classical and new.

In this talk I shall review this construction together with some of
its many applications, including some new results on complements of
subgroups of the free group, which are joint work with Pedro V. Silva.