  "Causal Dynamical Triangulations" (CDT) represent a lattice regularization of
the sum over spacetime histories, providing us with a non-perturbative
formulation of quantum gravity. The ultraviolet fixed points of the lattice
theory can be used to define a continuum quantum field theory, potentially
making contact with quantum gravity defined via asymptotic safety. We describe
the formalism of CDT, its phase diagram, and the quantum geometries emerging
from it. We also argue that the formalism should be able to describe a more
general class of quantum-gravitational models of Horava-Lifshitz type.
