The Grothendieck - Teichmuller Lie algebra grt is a Lie algebra, over the rational numbers Q, which is clearly very interesting and equally clearly not very well-understood.  It crops up in many different  areas of mathematics.  In this talk I will explain how the Lie algebra grt is related to the space of homotopy equivalences of the configuration spaces F(k, R^n) and F(k, S^n).  Here F(k,X) is the space of k distinct ordered points in the topological space X.  Much of the talk will be devoted to explaining, in as concrete a way as possible, the background needed for the statements of the results.  I will also try to explain one or two of the key ideas in the proofs.