  A Hermitian metric on a complex manifold of complex dimension $n$ is called
{\em astheno-K\"ahler} if its fundamental $2$-form $F$ satisfies the condition
$\partial \overline \partial F^{n - 2} =0$. If $n =3$, then the metric is {\em
strong KT}, i.e. $F$ is $\partial \overline \partial$-closed. By using blow-ups
and the twist construction, we construct simply-connected astheno-K\"ahler
manifolds of complex dimension $n > 3$. Moreover, we construct a family of
astheno-K\"ahler (non strong KT) $2$-step nilmanifolds of complex dimension $4$
and we study deformations of strong KT structures on nilmanifolds of complex
dimension $3$. Finally, we study the relation between astheno-K\"ahler
condition and (locally) conformally balanced one and we provide examples of
locally conformally balanced astheno-K\"ahler metrics on $\T^2$-bundles over
(non-K\"ahler) homogeneous complex surfaces.
