TAbstract. J. Nielsen showed early in the 1900's how to convert a pair of words u,v generating a free group F on x,y, to x,y via a simple algorithm. Andrews and Curtis conjectured (in the 1960's) that by augmenting Nielsen's transformations by conjugation, one obtains a procedure for converting any annihilating pair r, s (i.e. with normal closure all of F) to x, y. This has applications in topology, specifically to homotopy spheres. The conjecture will be described in detail, and examples considered. The talk will be elementary.