  We introduce two notions of hyperbolicity for not necessarily K\"ahler even
balanced $n$-dimensional compact complex manifolds $X$. The first, called {\it
SKT hyperbolicity}, generalises Gromov's K\"ahler hyperbolicity by means of SKT
metrics. The second, called {\it Gauduchon hyperbolicity} by means of Gauduchon
metrics. Our first main result in this paper asserts that every SKT hyperbolic
$X$ is also Kobayashi/Brody hyperbolic and every Gauduchon hyperbolic $X$ is
divisorially hyperbolic. The second main result is to prove a vanishing theorem
for the $L^2$ harmonic spaces on the universal cover of a SKT hyperbolic
manifold.
