  Let $\mathfrak{M}(\Sigma)$ be an open and connected subset of the space of
hyperbolic metrics on a closed orientable surface, and $\mathfrak{M}(M)$ an
open and connected subset of the space of metrics on an orientable manifold of
dimension at least $3$. We impose conditions on $M$ and $\mathfrak{M}(M)$,
which are often satisfied when the metrics in $\mathfrak{M}(M)$ have
non-positive curvature. Under these conditions, the data of a homotopy class of
maps from $\Sigma$ to $M$ gives $\mathfrak{M}(\Sigma)\times \mathfrak{M}(M)$
the structure of a space of harmonic maps. Using transversality theory for
Banach manifolds, we prove that the set of somewhere injective harmonic maps is
open, dense, and connected in the moduli space. We also prove some results
concerning the distribution of harmonic immersions and embeddings in the moduli
space.
