Quasi one-dimensional systems may result due to strong anisotropies in properties of a given higher-dimensional material, specifically in their transport properties. It turns out that some one-dimensional systems are exactly solvable, at least some of their properties may be obtained exactly. Typically, the energy spectrum is obtained exactly, and often their thermodynamic properties but general correlation functions may be hard to obtain, particularly dynamical correlation functions. Here we consider a model that is solvable, the Luttinger model, introducing a mapping between the electronic degrees of freedom and some collective bosonic degrees of freedom, in a process called bosonization. An interesting consequence is the separation of charge and spin degrees of freedom inside the one-dimensional system. Another method that allows exact solutions is the Bethe ansatz, and the solution of the isotropic Heisenberg spin-1/2 chain is presented in some detail. Other interesting problems are polyacetylene with soliton like solutions and the existence of unusual charge-spin relations and the case of a spin-1 chain that shows topological properties.