  We investigate instantons on sine-cones over Sasaki-Einstein and 3-Sasakian
manifolds. It is shown that these conical Einstein manifolds are K"ahler with
torsion (KT) manifolds admitting Hermitian connections with totally
antisymmetric torsion. Furthermore, a deformation of the metric on the
sine-cone over 3-Sasakian manifolds allows one to introduce a hyper-K"ahler
with torsion (HKT) structure. In the large-volume limit these KT and HKT spaces
become Calabi-Yau and hyper-K"ahler conifolds, respectively. We construct gauge
connections on complex vector bundles over conical KT and HKT manifolds which
solve the instanton equations for Yang-Mills fields in higher dimensions.
