  A Hermitian structure on a manifold is called locally conformally K\"ahler
(LCK) if it locally admits a conformal change which is K\"ahler. In this survey
we review recent results of invariant LCK structures on solvmanifolds and
present original results regarding the canonical bundle of solvmanifolds
equipped with a Vaisman structure, that is, a LCK structure whose associated
Lee form is parallel.
