  In the quest of a physical theory of quantum gravity, spin foam models, or in
short spinfoams, propose a well-defined path integral summing over quantized
discrete space-time geometries. At the crossroad of topological quantum field
theory, dynamical triangulations, Regge calculus, and loop quantum gravity,
this framework provides a non-perturbative and background independent
quantization of general relativity. It defines transition amplitudes between
quantum states of geometry, and gives a precise picture of the Planck scale
geometry with quantized areas and volumes. Gravity in three space-time
dimensions is exactly quantized in terms of the Ponzano-Regge state-sum and
Turaev-Viro topological invariants. In four space-time dimensions, gravity is
formulated as a topological theory, of the BF type, with extra constraints, and
hence quantized as a topological state-sum filled with defects. This leads to
the Engle-Pereira-Rovelli-Livine (EPRL) spinfoam model, that can be used for
explicit quantum gravity computations, for example for resolving the Big Bang
singularity by a bounce or in black-to-white hole transition probability
amplitudes.
