  We study the existence of lattices in almost abelian Lie groups that admit
left invariant locally conformal K\"ahler or locally conformal symplectic
structures in order to obtain compact solvmanifolds equipped with these
geometric structures. In the former case, we show that such lattices exist only
in dimension $4$, while in the latter case we provide examples of such Lie
groups admitting lattices in any even dimension.
