The spinfoam framework is a proposal for a regularized path integral for quantum gravity. Spinfoams define quantum space-time structures describing the evolution in time of the spin network states for quantum geometry derived from Loop Quantum Gravity (LQG). The construction of this covariant approach is based on the formulation of General Relativity as a topological theory plus the so-called simplicity constraints which introduce local degrees of freedom. The simplicity constraints are essential in turning the non-physical topological theory into 4d gravity. In this PhD manuscript, an original way to impose the simplicity constraints in 4d Euclidean gravity using harmonic oscillators is proposed and new coherent states, solutions of the constraints, are given. Moreover, a consistent spinfoam model for quantum gravity has to be connected to LQG and must have the right semi-classical limit. An explicit map between the spin network states of LQG and the boundary states of spinfoam models is given connecting the canonical and the covariant approaches. Finally, new techniques to compute semiclassical asymptotic expressions for the transition amplitudes of 3d quantum gravity and to extract semi-classical information from a spinfoam model are introduced. Explicit computations based on approximation methods and on the use of recurrence relations on spinfoam amplitudes have been performed. The results are relevant to derive quantum corrections to the dynamics of the gravitational field.Comment: PhD Thesis; Ecole Normale Sup\'erieure de Lyon. (192pages, many figures
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