  The moduli space of N=(4,4) string theories with a K3 target space is
determined, establishing in particular that the discrete symmetry group is the
full integral orthogonal group of an even unimodular lattice of signature
(4,20). The method combines an analysis of the classical theory of K3 moduli
spaces with mirror symmetry. A description of the moduli space is also
presented from the viewpoint of quantum geometry, and consequences are drawn
concerning mirror symmetry for algebraic K3 surfaces.
