The goal of statistical physics is to understand the properties of macroscopic systems from the interactions between their microscopic constituents. The first vertex model was introduced in this context, to study the thermodynamic properties of ice. Over the years, vertex models have been shown to exhibit rich analytic and algebraic structures, including surprising links to combinatorics. In this dissertation, we study the eight-vertex model and related systems such as the dynamical six-vertex (or eight-vertex SOS) model, the XYZ spin chain, as well as their higher-spin generalisations. The first part of this thesis discusses the periodic eight-vertex model. For a particular subfamily of its parameters and an odd system size, its transfer matrix possesses a remarkably simple eigenvalue. Our goal is to investigate the corresponding eigenspace. We compute several scalar products and components of the basis vectors in terms of special polynomials introduced by Rosengren and Zinn-Justin. The proofs of these results rely on a well-established induction technique due to Izergin and Korepin. The second part of this thesis discusses the quasi-periodic forty-one-vertex model, a spin-1 generalisation of the eight-vertex model. We establish that its transfer matrix possesses a remarkably simple eigenvalue, for any system size or parameter values. Our goal is to investigate the corresponding eigenspace. We provide a characterisation of the eigenvector. Moreover, we compute a quadratic sum rule involving it in terms of new special polynomials connected to Rosengren and Zinn-Justin's. The proof of these results centres around the use of quantum Separation of Variables to solve the eigenvalue problem of a related dynamical ten-vertex model. (SC - Sciences) -- UCL, 2024