  New relations involving the Riemann, Ricci and Einstein tensors that have to
hold for a given geometry to admit Killing-Yano tensors are described. These
relations are then used to introduce novel conserved "currents" involving such
Killing-Yano tensors. For a particular current based on the Einstein tensor, we
discuss the issue of conserved charges and consider implications for matter
coupled to gravity. The condition on the background geometry to allow
asymptotic conserved charges for a current introduced by Kastor and Traschen is
found and a number of other new aspects of this current are commented on. In
particular we show that it vanishes for rank $(D-1)$ Killing-Yano tensors in
$D$ dimensions.
