Introduction to Dubois-Violette's noncommutative differential geometry

We present a detailed review of the Dubois-Violette approach to noncommutative differential calculus. The noncommutative differential geometry of matrix algebras and the noncommutative Poisson structures are treated in some detail. We also present the analog of Maxwell's theory and new models of Yang-Mills-Higgs theories that can be constructed in this framework. In particular, some simple models are compared with the standard model. Finally, we discuss some perspectives and open questions.