  We study sets of solutions to equations over a free group, projections of
such sets, and the structure of elementary sets defined over a free group. The
structre theory we obtain enable us to answer some questions of A. Tarski's,
and classify those finitely generated groups that are elementary equivalent to
a free group. Connections with low dimensional topology, and a generalization
to (Gromov) hyperbolic groups will also be discussed.
