  In this thesis I review the definition of topological quantum field theories
through state sums on triangulated manifolds. I describe the construction of
state sum invariants of 3-manifolds from a graphical calculus and show how to
evaluate the invariants as boundary amplitudes. I review how to define such a
graphical calculus through SU(2) representation theory. I then review various
geometricity results for the representation theory of SU(2), Spin(4) and
SL(2,C), and define coherent boundary manifolds for state sums based on these
representations. I derive the asymptotic geometry of the SU(2) based
Ponzano-Regge invariant in three dimensions, and the SU(2) based Ooguri models
amplitude in four dimensions. As a corollary to the latter results I derive the
asymptotic behaviour of various recently proposed spin foam models motivated
from the Plebanski formulation of general relativity. Finally the asymptotic
geometry of the SL(2,C) based model is derived.
