  We define reduction of locally conformal Kaehler manifolds, considered as
conformal Hermitian manifolds, and we show its equivalence with an unpublished
construction given by Biquard and Gauduchon. We show the compatibility between
this reduction and Kaehler reduction of the universal cover. By a recent result
of Kamishima and the second author, in the Vaisman case (that is, when a metric
in the conformal class has parallel Lee form) if the manifold is compact its
universal cover comes equipped with the structure of Kaehler cone over a Sasaki
compact manifold. We show the compatibility between our reduction and Sasaki
reduction, hence describing a subgroup of automorphisms whose action causes
reduction to bear a Vaisman structure. Then we apply this theory to construct a
wide class of Vaisman manifolds.
