  This thesis is devoted to the study of 3-dimensional quantum gravity as a
spin foam model and group field theory. In the first part of this thesis, we
review some general physical and mathematical aspects of 3-dimensional gravity,
focusing on its topological nature. Afterwards, we review some important
aspects of the Ponzano-Regge spin foam model for 3-dimensional Riemannian
quantum gravity and explain in some details how it is related to the
discretized path integral of general relativity in its first-order formulation.
Furthermore, we discuss briefly some related spin foam models and review the
notion of spin network states in order to properly define transition amplitudes
of these models.
  The main results of this thesis are contained in the second part. We start by
reviewing the Boulatov group field theory and explain how it is related to the
Ponzano-Regge model and some advantages of introducing colouring. Afterwards,
we give a very detailed review of the topology of coloured graphs with
non-empty boundary and review techniques, which are devolved in crystallization
theory, a branch of geometric topology. In the last part of this chapter, we
apply these techniques in order to define suitable boundary observables and
transition amplitudes of this model and in order to set up a formalism for
dealing with transition amplitudes in the coloured Boulatov model in a more
systematic way by writing them as topological expansions. We also apply these
techniques to the simplest possible boundary state representing a 2-sphere.
  Last but not least, we review some results regarding quasi-local holography
in the Ponzano-Regge model, construct some explicit examples of coloured graphs
representing manifolds with torus boundary and discuss the transition amplitude
of some fixed boundary graph representing a 2-torus.
