  We introduce two notions of hyperbolicity for not necessarily K\"ahler
$n$-dimensional compact complex manifolds $X$. The first, called {\it balanced
hyperbolicity}, generalises Gromov's K\"ahler hyperbolicity by means of
Gauduchon's balanced metrics. The second, called {\it divisorial
hyperbolicity}, generalises the Brody hyperbolicity by ruling out the existence
of non-degenerate holomorphic maps from $\C^{n-1}$ to $X$ that have what we
term a subexponential growth. Our main result in the first part of the paper
asserts that every balanced hyperbolic $X$ is also divisorially hyperbolic. We
provide a certain number of examples and counter-examples and discuss various
properties of these manifolds. In the second part of the paper, we introduce
the notions of {\it divisorially K\"ahler} and {\it divisorially nef} real De
Rham cohomology classes of degree $2$ and study their properties. They also
apply to $C^\infty$, not necessarily holomorphic, complex line bundles and are
expected to be implied in certain cases by the hyperbolicity properties
introduced in the first part of the work. While motivated by the observation of
hyperbolicity properties of certain non-K\"ahler manifolds, all these four new
notions seem to have a role to play even in the K\"ahler and the projective
settings.
