A {\em connectivity function on} a set $E$ is a function $��:2^E\rightarrow \mathbb R$ such that $��(\emptyset)=0$, that $��(X)=��(E-X)$ for all $X\subseteq E$ and that $��(X\cap Y)+��(X\cup Y)\leq ��(X)+��(Y)$ for all $X,Y \subseteq E$. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. We introduce a notion of duality for polymatroids and prove that every connectivity function is the connectivity function of a self-dual polymatroid. We also prove that every integral connectivity function is the connectivity function of a half-integral self-dual polymatroid.