The Luttinger model: its role in the RG--theory of one--dimensional many body Femi systems

The Luttinger model was introduced to illustrate the possibility of a perturbative treatment of the singularity at the Fermi surface, already known to be ``anomalous'' from the results of the theory of Tomonaga, via an exactly soluble model. It became soon the subject of great interest also on the part of Mathematical Physics and a key to the investigations of the mathematical properties of Condensed Matter Physics. This paper reviews aspects of the above developments relevant for renormalization group methods by illustrating the conceptual development of the renormalization group approach to the ground state theory of the $1$--dimensional spinless Fermi gas at small coupling.