One of the goals of statistical physics is to understand the magnetic properties of the matter from the atomic properties and interactions. In this dissertation, we study the famous XXZ and XYZ spin-chains, which are one-dimensional spin-1/2 models. For specific values of their parameters, these spin chains can be expressed in the framework of supersymmetric quantum mechanics. This supersymmetric structure implies that the Hamiltonians of these models may possess special ground states called supersymmetry singlets. The first objective of this thesis is to exploit a relation between the supersymmetry and the theory of (co)homology to detect the existence and characterise supersymmetry singlets. Furthermore, we use the supersymmetry of XYZ spin chains with various boundary conditions to prove the existence of remarkably simple eigenvalues of the transfer matrices of the related supersymmetric eight-vertex models. The second objective of this thesis is to introduce the multipartite fidelity, a measure of the quantum entanglement for systems with a spatial extension, and to compute this quantity for the XXZ spin chain by using the supersymmetry. Special components and linear sum rules of XXZ and XYZ spin chains, for specific value of their parameters, are related to combinatorics and in particular to the enumeration of alternating sign matrices (ASMs). The third objective of this thesis is to use the integrability to gain new insight into the combinatorial properties of the spin-chain ground states. Namely, we find a solution to the boundary quantum Knizhnik-Zamolodchikov equations and use it to prove formulas for special components of the open XXZ supersymmetry singlet related to ASMs. (SC - Sciences) -- UCL, 2020