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We briefly review ideas about ``noncommutativity of space-time'' and approaches toward a corresponding theory of gravity.
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A. Dimakis and F. Müller-Hoissen, Discrete differential calculus, graphs, topologies and gauge theory, J. Math. Phys. 35 (1994) 6703–6735
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S. Dopplicher, K. Fredenhagen, and J. E. Roberts, The quantum structure of spacetime at the Planck scale and quantum fields, Commun. Math. Phys. 172 (1995) 187–220
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A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (1995) 6194–6231
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M.A. Cardella and D. Zanon, Noncommutative deformation of four-dimensional Einstein gravity, Class. Quantum Grav. 20 (2003) L95–L103
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D.V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rep. 388 (2003) 279–360
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V. Moretti, Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes, Rev. Math. Phys. 15 (2003) 1171–1217
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A. Dimakis and F. Müller-Hoissen, Differential geometry of group lattices, J. Math. Phys. 44 (2003) 1781–1821
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A. Dimakis and F. Müller-Hoissen, Riemannian geometry of bicovariant group lattices, J. Math. Phys. 44 (2003) 4220–4259
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H. Baehr, A. Dimakis, and F. Müller-Hoissen, Differential calculi on commutative algebras, J. Phys. A: Math. Gen. 28 (1995) 3197–3222
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T. Filk, Divergencies in a field theory on quantum space, Phys. Lett. B 376 (1996) 53–58
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A. Dimakis and C. Tzanakis, Non-commutative geometry and kinetic theory of open systems, J.Phys. A 29 (1996) 577–594
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G. Landi, An Introduction to Noncommutative Spaces and their Geometries
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A. Klimyk and K. Schmüdgen, Quantum Groups and their Representations
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A.H. Chamseddine and A. Connes, The spectral action principle, Commun. Math. Phys. 186 (1997) 731–750
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H.-J. Matschull and M. Welling, Quantum mechanics of a point particle in ( 2 + 1 ) (2+1) -dimensional gravity, Class. Quantum Grav. 15 (1998) 2981–3030
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M. Chaichian, P.P. Kulish, K. Nishijima, and A. Tureanu, On a Lorentz-invariant interpretation of noncommutative space–time and its implications on noncommutative QFT, Phys. Lett. B 604 (2004) 98–102
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M. Chaichian, P. Prešnajder, and A. Tureanu, New concept of relativistic invariance in noncommutative space-time: twisted Poincaré symmetry and its implications, Phys. Rev. Lett. 94 (2005) 151602
2005
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P. Aschieri, C. Blohmann, M. Dimitrijević, F. Meyer, P. Schupp, and J. Wess, A gravity theory on noncommutative spaces, Class. Quantum Grav. 22 (2005) 3511–3532
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X. Calmet and A. Kobakhidze, Noncommutative general relativity, Phys. Rev. D 72 (2005) 045010
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S. Carlip, Quantum gravity in 2 + 1 2+1 dimensions: The case of a closed universe, Living Rev. Relativity 8 (2005) 1–63
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R.J. Szabo, Symmetry, gravity and noncommutativity, Class. Quantum Grav. 23 (2006) R199–R242
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J. Wess, Einstein-Riemann gravity on deformed spaces, SIGMA 2 (2006) 089
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L. Álvarez-Gaumé, F. Meyer, and M.A. Vázquez-Mozo, Comments on noncommutative gravity, Nucl. Phys. B 753 (2006) 92–117
2006
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X. Calmet and A. Kobakhidze, Second order noncommutative corrections to gravity, Phys. Rev. D 74 (2006) 047702
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J.R. Klauder, Overview of Affine Quantum Gravity, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 81–94
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A. Strohmaier, On noncommutative and pseudo-Riemannian geometry, J. Geom. Phys. 56 (2006) 175–195
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B.M. Zupnik, Reality in noncommutative gravity, Class. Quantum Grav. 24 (2007) 15–26
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A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives
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J.-H. Jureit, T. Krajewski, T. Schücker, and C.A. Stephan, On the noncommutative standard model, Acta Phys. Polon. B 38 (2007) 3181–3202
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M. Paschke, An essay on the spectral action and its relation to quantum gravity, in: Quantum Gravity, Mathematical Models and Experimental Bounds
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S. Zakrzewski, Quantum Poincaré group related to the κ \kappa -Poincaré algebra, J. Phys. A: Math. Gen. 27 (1994) 2075–2082
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