  We study the existence of strong K\"ahler with torsion (SKT) metrics and of
symplectic forms taming invariant complex structures $J$ on solvmanifolds
$G/\Gamma$ providing some negative results for some classes of solvmanifolds.
In particular, we show that if either $J$ is invariant under the action of a
nilpotent complement of the nilradical of $G$ or $J$ is abelian or $G$ is
almost abelian (not of type (I)), then the solvmanifold $G/\Gamma$ cannot admit
any symplectic form taming the complex structure $J$, unless $G/\Gamma$ is
K\"ahler. As a consequence, we show that the family of non-K\"ahler complex
manifolds constructed by Oeljeklaus and Toma cannot admit any symplectic form
taming the complex structure.
