  This thesis is broadly split into two parts. In the first part, simple state
sum models for minimally coupled fermion and scalar fields are constructed on a
$1$-manifold. The models are independent of the triangulation and give the same
result as the continuum partition functions evaluated using zeta-function
regularisation. Some implications for more physical models are discussed.
  In the second part, the gauge gravity action is written using a particularly
simple matrix technique. The coupling to scalar, fermion and Yang-Mills fields
is reviewed, with some small additions. A sum over histories quantisation of
the gauge gravity theory in 2+1 dimensions is then carried out for a particular
class of triangulations of the three-sphere. The preliminary stage of the
Hamiltonian analysis for the (3+1)-dimensional gauge gravity theory is
undertaken.
