Noncommutative geometry, in its many incarnations, appears at the crossroad
of various researches in theoretical and mathematical physics: from models of
quantum space-time (with or without breaking of Lorentz symmetry) to loop
gravity and string theory, from early considerations on UV-divergencies in
quantum field theory to recent models of gauge theories on noncommutative
spacetime, from Connes description of the standard model of elementary
particles to recent Pati-Salam like extensions. We list several of these
applications, emphasizing also the original point of view brought by
noncommutative geometry on the nature of time.
This text serves as an introduction to the volume of proceedings of the
parallel session "Noncommutative geometry and quantum gravity", as a part of
the conference "Conceptual and technical challenges in quantum gravity"
organized at the University of Rome "La Sapienza" in September 2014
