In this chapter we shall introduce the one-dimensional many-body quantum spin Hamiltonians and present some exact results for thermodynamic characteristics of the systems, described by those Hamiltonians. Such spin systems describe electron insulators in which charge degrees of freedom of electrons are frozen ( e.g., electrons are localized) and the only spin, magnetic excitations determine the states of electron systems. The simplest Hamiltonian of a spin system is the Zeeman Hamiltonian which describes the behaviour of spins in an external magnetic field. It shows how the degeneracy in the determination of the directions of spins is lifted by the magnetic field. However, one cannot describe the behaviour of most of magnetic insulators with only the Zeeman interaction. Generally speaking, there exists an interaction between spin degrees of motion of electrons in a crystal. It was concluded that the most important interaction, responsible for magnetic properties of multi-electron systems is the exchange interaction, which stems from the Coulomb interaction of electrons. The Heisenberg model is the seminal model of quantum mechanics. It was introduced to describe the exchange interaction of localized spin moments in insulators. It is commonly accepted that for most of the properties it is enough to consider exchange coupling between only nearest neighbours on the lattice. The reader already saw such a Hamiltonian in the previous chapter, when we considered the Mermin Wagner theorem. In the case of a three-dimensional lattice, the Heisenberg model successfully describes ferromagnetic or antiferromagnetic ordering with the help of a mean-field-like...