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We study the existence of strong K\"ahler with torsion (SKT) metrics and ofsymplectic 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 anilpotent complement of the nilradical of $G$ or $J$ is abelian or $G$ isalmost abelian (not of type (I)), then the solvmanifold $G/\Gamma$ cannot admitany symplectic form taming the complex structure $J$, unless $G/\Gamma$ isK\"ahler. As a consequence, we show that the family of non-K\"ahler complexmanifolds constructed by Oeljeklaus and Toma cannot admit any symplectic formtaming the complex structure.

SKT and Tamed Symplectic structures on solvmanifolds