  We study HKT structures on nilpotent Lie groups and on associated
nilmanifolds. We exhibit three weak HKT structures on $\R^8$ 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 ${\frak sp}(n)$ to
produce continuous families of compact and noncompact of manifolds carrying non
isometric HKT structures. Finally, geometrical properties of invariant HKT
structures on 2-step nilpotent Lie groups are obtained.
