Summary: We study hyper-Kähler torsion (HKT) structures on nilpotent Lie groups and on associated nilmanifolds. We show three weak HKT structures on \(\mathbb{R}^8\) which are homogeneous with respect to extensions of Heisenberg-type Lie groups. The corresponding hypercomplex structures are of a special kind called Abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from Abelian hypercomplex structures. Furthermore, we use a correspondence between Abelian hypercomplex structures and subspaces of \({\mathfrak s}{\mathfrak p}(n)\) to produce continuous families of compact and noncompact manifolds carrying non-isometric HKT structures. Finally, geometrical properties of invariant HKT structures on 2-step nilpotent Lie groups are obtained.