When the domain is a surface $M$ of the conformal type of the complex plane, and the target is a Hadamard surface $N$, a result of myself with collaborators Tam, Treibergs, and Wan characerizes harmonic diffeomorphisms from $M$ into $N$ with polynomial growth as harmonic maps with polynomial quadratic differentials and $M$ is mapped, under such maps, onto finite-sided ideal polygons in $N$. This result has been partially extended later by P. Li and J. Wang to higher dimensional setting.
One interesting aspect of the $2-$dimensional work is the close interplay between several subjects: the conformal invarance of the harmonic map equation when the domain is a surface and the relevant, invariant, geometric PDE estimates, the geometry of the foliations of the holomorphic quadratic differential associated with a harminic map from surfaces, and the Gauss map of complete spacelike constant mean curvature (SCMC) surfaces in the Minkowski space ${\bf M^{2,1}}$. The connection with the last occurs when $N$ is the hyperbolic plane $\mathbb H^2$, in which case harmonic maps into $\mathbb H^2$ can be locally represented as the Gauss map of SCMC surfaces in the Minkowski space ${\bf M^{2,1}}$. The SCMC representations of harmonic diffeomorphisms into $\mathbb H^2$ with polynomial growth corresponds to complete SCMC surfaces in the Minkowski space ${\bf M^{2,1}}$ with finite total curvature. Here, there is also an interesting contrast with the rich and beautiful theory of the distribution of the Gauss map of minimal surfaces in Euclidean 3 space, in particular those with total finite curvature.
There are many unanswered questions in this area. For instance, it is natural to ask whether one can determine the conformal type of a surface through the behavior at $\infty$ of harmonic maps from it to the hyperbolic plane? Can one obtain some kind of ``value distribution theory", or even some quantitative ``defect relation" on the points at $\infty$ of the hyperbolic plane which are in the closure of the image of the surface under a harmonic map? More generally, one could ask: how to describe the geometry of a harmonic map from a surface into a negatively curved surface/space? Are there any structure/restriction on the limit set at $\infty$ of the image of the harmonic map?