This paper deals with Hamiltonian realizations for Hamiltonian transfer functions \(G(Z)=G^ T(-Z)\). After the introduction, Chapter 2 develops the theory of duality for polynomial models in symplectic spaces. This leads to an intimate study of Hamiltonian adjoints of Laurent maps. In Chapter 3, Hamiltonian realizations for Hamiltonian transfer functions are developed. Here a triple of maps A, B, C is called a Hamiltonian system in a symplectic space X if it is skew-symmetric in the alternating metric of X, i.e. \(A^ H=-A\), \(B^ H=C\), \(C^ H=B\). In order to develop a Hamiltonian realization from a coprime realization of such transfer functions, a polynomial Hamiltonian equivalence relation is introduced and representatives for each equivalence class are found localizing at the singularities of the transfer function. This development takes about 60 \% of the paper and hinges on the results of chapter 2 and a string of lemmas on the eigenvalue or inertia structure in a symplectic space of 2n\(\times 2n\) block matrices of the type \(\left( \begin{matrix} A\quad B\\ -B\quad -A\end{matrix} \right)\) or related types. These matrix theoretical results would have deserved to be published separately. Here they are lost among the various cases studied for singularities of the Hamiltonian transfer function. The last chapter applies the knowledge gained above about a skew- symmetric map jointly with a symmetric map to classify products of skew symmetric and symmetric matrices and to find a canonical form under congruence of such pencils. These last results are not new. Questions like these go back to Kronecker, Weierstraß and Frobenius. They were first solved by Williamson, Wall et al. The bibliography is long, explicit and nicely integrated into the beautiful paper.