  We determine the 6-dimensional solvmanifolds admitting an invariant complex
structure with holomorphically trivial canonical bundle. Such complex
structures are classified up to isomorphism, and the existence of strong
K\"ahler with torsion (SKT), generalized Gauduchon, balanced and strongly
Gauduchon metrics is studied. As an application we construct a holomorphic
family $(M,J_a)$ of compact complex manifolds such that $(M,J_a)$ satisfies the
$\partial\bar\partial$-Lemma and admits a balanced metric for any $a\not=0$,
but the central limit neither satisfies the $\partial\bar\partial$-Lemma nor
admits balanced metrics.
