  One can try to define the theory of quantum gravity as the sum over
geometries. In two dimensions the sum over {\it Euclidean} geometries can be
performed constructively by the method of {\it dynamical triangulations}. One
can define a {\it proper-time} propagator. This propagator can be used to
calculate generalized Hartle-Hawking amplitudes and it can be used to
understand the the fractal structure of {\it quantum geometry}. In higher
dimensions the philosophy of defining the quantum theory, starting from a sum
over Euclidean geometries, regularized by a reparametrization invariant cut off
which is taken to zero, seems not to lead to an interesting continuum theory.
The reason for this is the dominance of singular Euclidean geometries.
Lorentzian geometries with a global causal structure are less singular. Using
the framework of dynamical triangulations it is possible to give a constructive
definition of the sum over such geometries, In two dimensions the theory can be
solved analytically. It differs from two-dimensional Euclidean quantum gravity,
and the relation between the two theories can be understood. In three
dimensions the theory avoids the pathologies of three-dimensional Euclidean
quantum gravity. General properties of the four-dimensional discretized theory
have been established, but a detailed study of the continuum limit in the
spirit of the renormalization group and {\it asymptotic safety} is till
awaiting.
