The abelian group $\Pext GH$ of pure extensions has recently attracted the interest of workers in non-commutative topology, especially those using $KK$-theory, since under minimal hypotheses the closure of zero in the Kasparov group $KK_*(A,B)$ (for separable $C^*$-algebras $A$ and $B$) is isomorphic to the group \[ \Pext{K_*(A)}{K_*(B)}. \] As $K_*(A)$ and $K_*(B)$ can take values in all countable abelian groups, assuming that $G$ and $H$ are countable is natural.

In this mostly expository work we survey the known (and not so well-known) properties of $\pext$ and its relationship to $\lim ^1 $ and develop some new results on their computation.