  The aim of this article is to study the existence of invariant SKT structures
on nilmanifolds. More precisely, we give a negative answer to the question of
whether there exist a $k$-step ($k>2$) complex nilmanifold admitting an
invariant SKT metric. We also provide a construction which serves as a tool to
generate examples of invariant SKT structures on $2$-step nilmanifolds in
arbitrary dimensions.
