  We study left-invariant locally conformally K\"ahler structures on Lie
groups, or equivalently, on Lie algebras. We give some properties of these
structures in general, and then we consider the special cases when its complex
structure is bi-invariant or abelian. In the former case, we show that no such
Lie algebra is unimodular, while in the latter, we prove that if the Lie
algebra is unimodular, then it is isomorphic to the product of $\mathbb R$ and
a Heisenberg Lie algebra.
