This paper provides a detailed study of symmetric transfer functions. First, an introduction to polynomial models and their use in realization theory as well as a discussion on bases and dual bases in the context of polynomial models are given. Then the class of multivariable or generalized Bezoutians is studied in the same context and the polynomial models are used to rederive the scalar Hermite-Hurwitz theorem. After that a simplified treatment of the Frobenius result for computing the signature of a Hankel matrix is made through the use of Bezoutians which is more directly related the Euclidean algorithm. Then the author passes to the explicit construction of an appropriate signature symmetric realization of a scalar transfer function \(g=p/q\) which may be of the Rosenbrock type \(g=(re)q^{-1}r+s\), where the signature information is carried in the polynomial e. The paper continues with a proof of a multivariable generalization of the Chinese remainder theorem, and a study of partial fraction decompositions with matrix fractions where special attention is given to the implications of the symmetry property of the rational function. Using the preceding tools, the author proceeds to the construction of signature symmetric realizations of real symmetric rational transfer functions. The paper closes with an application of polynomial models in the proof of a generalized version of a theorem of Frobenius, and in the study of self- adjoint operators in indefinite metric spaces, and especially their reduction to canonical form under the group of orthogonal matrices in this metric. The methods studied and employed in the paper depend strongly on a bilinear form defined on the space of truncated vector Laurent series. The case of real rational transfer functions possessing Hamiltonian symmetry, and of canonical forms for Hamiltonian maps is proposed by the author for further study. We think that many of the results presented in this paper can be carried over to the 2-dimensional case which covers 2- dimensional filters and systems.