  We prove the existence of nonconstant harmonic maps of optimal regularity
from an arbitrary closed manifold $(M^n,g)$ of dimension $n>2$ to any closed,
non-aspherical manifold $N$ containing no stable minimal two-spheres. In
particular, this gives the first general existence result for harmonic maps
from higher-dimensional manifolds to a large class of positively curved
targets. In the special case of the round spheres $N=\mathbb{S}^k$, $k\geq 3$,
we obtain a distinguished family of nonconstant harmonic maps $M\to
\mathbb{S}^k$ of index at most $k+1$, with singular set of codimension at least
$7$ for $k$ sufficiently large. Furthermore, if $3\leq n\leq 5$, we show that
these smooth harmonic maps stabilize as $k$ becomes large, and correspond to
the solutions of an eigenvalue optimization problem on $M$, generalizing the
conformal maximization of the first Laplace eigenvalue on surfaces.
