Classical Spanier-Whitehead duality was introduced for the stable homotopy category of finite CW complexes. We consider 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 Kasparov $KK$-groups. Examples from foliations, hyperbolic dynamics, and other highly non-commutative $C^*$-algebras illustrate the truly new phenomena encountered. There are many open questions associated with relaxing the assumptions on the algebras. For example, does the Calkin algebra have a Spanier-Whitehead $K$-dual? This is joint work with JerryKaminker.