  Non-commutative geometry (NCG) is a mathematical discipline developed in the
1990s by Alain Connes. It is presented as a new generalization of usual
geometry, both encompassing and going beyond the Riemannian framework, within a
purely algebraic formalism. Like Riemannian geometry, NCG also has links with
physics. Indeed, NCG provided a powerful framework for the reformulation of the
Standard Model of Particle Physics (SMPP), taking into account General
Relativity, in a single "geometric" representation, based on Non-Commutative
Gauge Theories (NCGFT). Moreover, this accomplishment provides a convenient
framework to study various possibilities to go beyond the SMPP, such as Grand
Unified Theories (GUTs). This thesis intends to show an elegant method recently
developed by Thierry Masson and myself, which proposes a general scheme to
elaborate GUTs in the framework of NCGFTs. This concerns the study of NCGFTs
based on approximately finite $C^*$-algebras (AF-algebras), using either
derivations of the algebra or spectral triples to build up the underlying
differential structure of the Gauge Theory. The inductive sequence defining the
AF-algebra is used to allow the construction of a sequence of NCGFTs of
Yang-Mills Higgs types, so that the rank $n+1$ can represent a grand unified
theory of the rank $n$. The main advantage of this framework is that it
controls, using appropriate conditions, the interaction of the degrees of
freedom along the inductive sequence on the AF algebra. This suggests a way to
obtain GUT-like models while offering many directions of theoretical
investigation to go beyond the SMPP.
