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We review the present status of gauge theories built on various quantum space–times described by noncommutative space–times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with particular emphasis on some explored quantum properties. Recent attempts aiming to include gravity dynamics within a noncommutative framework are also considered. [ABSTRACT FROM AUTHOR]

κ -Poincaré invariant gauge theories on κ -Minkowski space-time, which are noncommutative analogs of the usual U(1) gauge theory, exist only in five dimensions. These are built from noncommutative twisted connections on a hermitian right module over the algebra coding the κ -Minkowski space-time. We show that twisting the action of this algebra on the hermitian module, assumed to be a copy of it, affects neither the value of the above dimension nor the noncommutative gauge group defined as the unitary automorphisms of the module leaving the hermitian structure unchanged. Only the hermiticity condition obeyed by the gauge potential becomes twisted. Similarities between the present framework and algebraic features of twisted spectral triples are exhibited. [ABSTRACT FROM AUTHOR]