  We introduce and study a notion of `Sasaki with torsion structure' (ST) as an
odd-dimensional analogue of K\"ahler with torsion geometry (KT). These are
normal almost contact metric manifolds that admit a unique compatible
connection with 3-form torsion. Any odd-dimensional compact Lie group is shown
to admit such a structure; in this case the structure is left-invariant and has
closed torsion form.
  We illustrate the relation between ST structures and other generalizations of
Sasaki geometry, and explain how some standard constructions in Sasaki geometry
can be adapted to this setting. In particular, we relate the ST structure to a
KT structure on the space of leaves, and show that both the cylinder and the
cone over an ST manifold are KT, although only the cylinder behaves well with
respect to closedness of the torsion form. Finally, we introduce a notion of
`G-moment map'. We provide criteria based on equivariant cohomology ensuring
the existence of these maps, and then apply them as a tool for reducing ST
structures.
