  Classical Spanier-Whitehead duality was introduced for the stable homotopy
category of finite CW complexes. Here we provide a comprehensive treatment of a
noncommutative version, termed Spanier-Whitehead $K$-duality, which is defined
on the category of $C^*$-algebras whose $K$-theory is finitely generated and
that satisfy the UCT with morphisms the $KK$-groups. We explore what happens
when these assumptions are relaxed in various ways. In particular, we consider
the relationship between Paschke duality and Spanier-Whitehead $K$-duality.
