  We investigate in detail the connection between harmonic maps from Riemann
surfaces into the unitary group $\U(n)$ and their Grassmannian models: these
are families of shift-invariant subspaces of $L^2(S^1,\C^n)$. With the help of
operator-theoretic methods we derive a criterion for finiteness of the uniton
number which has a large number of applications discussed in the paper.
