  I prove three classification results about harmonic morphisms whose fibers
have dimension one. All are valid when the domain is at least of dimension 4.
(The character of this overdetermined problem is very different when the
dimension of the domain is 3 or less.) The first result is a local
classification for such harmonic morphisms with specified target metric, the
second is a finiteness theorem for such harmonic morphisms with specified
domain metric, and the third is a complete classification of such harmonic
morphisms when the domain is a space form of constant sectional curvature
  The methods used are exterior differential systems and the moving frame. The
fundamental results are local, but, because of the rigidity of the solutions,
they allow a complete global classification.
