General Definition and Current Status of Quantum Space-Times in Quantum Gravity. The concept of quantum space-times (QST) is fundamental to the development of relativistic quantum theories and at this point it can only be broadly defined as a class of mathematical spaces that allow the construction of quantum physical theories in a manner consistent with both relativistic principles and quantum gravity. There is no universal agreement amongst either theoretical physicists or mathematicians who work on Physical Mathematics about either a specific definition of such quantum space-times or how to develop a valid classification theory of quantum space-times. However, several specific definitions or models were proposed and a list of such examples is presented next.

Proposed Specific Definitions or Models of Quantum Space-Times (QSTs)

• QSTs represented by posets or causal sets
• QSTs represented by so-called `quantum' topoi with Heyting logic algebra as classifier
• QSTs represented by Topological Quantum Field Theories (TQFTs) or Homotopy QFTs
• QSTs represented as spin foams of spin networks
• QSTs represented as a noncommutative, algebraic- and/or `geometrical'- quantized space as in noncommutative geometry of SUSY
• QSTs represented as generalized Riemannian manifolds with quantum tangent spaces
• QSTs represented as presheaves of local nets of quantum operators in Algebraic QFT
• QSTs represented as Quantum Fields on a (`physical') Lattice of geometric points
• QSTs represented as consisting of quantum `loops'
• QSTs represented as fractal dimension spaces
• QSTs represented as categories or spaces of `strings' as in `string theories'
• `Twistor' representations in Quantum Gravity (introduced by Sir Roger Penrose).