We study the existence of strong K��hler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures $J$ on solvmanifolds $G/��$ 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 $G/��$ cannot admit any symplectic form taming the complex structure $J$, unless $G/��$ is K��hler. As a consequence, we show that the family of non-K��hler complex manifolds constructed by Oeljeklaus and Toma cannot admit any symplectic form taming the complex structure.

Final version of the paper "Tamed complex structures on solvmanifolds". To appear in Tohoku J. Math