  I review the formalism of loop quantum gravity, in both its real and complex
formulations, and spin foam theory which is its path integral counterpart. Spin
networks for non-compact groups are introduced (following hep-th/0205268) to
deal with gauge invariant structures based on the Lorentz group. The whole
formalism is studied in details in three dimensions in both its canonical
formulation (loop gravity) and its spin foam formulation. The main output
(following gr-qc/0212077) is the discreteness of timelike intervals and the
continuous character of spacelike distances even at the quantum level. Then it
is explained how to extend these considerations to the 4-dimensional case. I
review the Barrett-Crane model, its geometrical interpretation, its link with
general relativity and the role of causality. It is shown to be the history
formulation of a covariant canonical formulation of loop gravity (following
gr-qc/0209105), whose link with standard loop quantum gravity is discussed.
Similarly to the 3d case, spacelike areas turn out continuous. Finally, ways of
extracting informations from the non-perturbative spin foam structures are
discussed.
