A structural description is given of the separable C*-algebras that are contractible in KK-theory or E-theory. Subsequently it is shown that two separable C*-algebras, A and B, are KK-equivalent if and only if there is a bundle of separable C*-algebras over [0,1] which is piecewise trivial with no more than six points of non-triviality such that the kernel of each fiber map is KK-contractible, all fiber maps are semi-split and such that the fibers at the endpoints of the interval are A and B.