We study harmonic morphisms from domains in ${\mathbf{R}}^3$ and $S^3$ to a Riemann surface $N$, obtaining the classification of such in terms of holomorphic mappings from a covering space of $N$ into certain Grassmannians. We show that the only non-constant submersive harmonic morphism defined on the whole of $S^3$ to a Riemann surface is essentially the Hopf map.Comparison is made with the theory of analytic functions. In particular we consider multiple-valued harmonic morphisms defined on domains in ${\mathbf{R}}^3$ and show how a cutting...