Combinatorial group theory has its roots in topology and in particular in the theory of the fundamental group. Introduced by Poincare in giving an
example of a 3-manifold which has trivial homology but is not homeomorphic to the 3-sphere, the fundamental group is a powerful if sometimes intractable invariant of a topological space. The aim of this essay is to describe the complex interaction between the algebra and the geometry that is transmitted through the medium of the fundamental group.