  We study complex solvmanifolds $\Gamma\backslash G$ with holomorphically
trivial canonical bundle. We show that the trivializing section of this bundle
can be either invariant or non-invariant by the action of $G$. First we
characterize the existence of invariant trivializing sections in terms of the
Koszul 1-form $\psi$ canonically associated to $(\mathfrak{g},J)$, where
$\mathfrak{g}$ is the Lie algebra of $G$, and we use this characterization to
produce new examples of complex solvmanifolds with trivial canonical bundle.
Moreover, we provide an algebraic obstruction, also in terms of $\psi$, for a
complex solvmanifold to have trivial (or more generally holomorphically
torsion) canonical bundle. Finally, we exhibit a compact hypercomplex
solvmanifold $(M^{4n},\{J_1,J_2,J_3\})$ such that the canonical bundle of
$(M,J_{\alpha})$ is trivial only for $\alpha=1$, so that $M$ is not an
$\operatorname{SL}(n,\mathbb{H})$-manifold.
