  To solve the path integral for quantum gravity, one needs to regularise the
space-times that are summed over. This regularisation usually is a
discretisation, which makes it necessary to give up some paradigms or
symmetries of continuum physics.
  Causal dynamical triangulations regularises the path integral through a
simplicial discretisation that introduces a preferred time foliation.
  The first part of this thesis presents three articles on causal dynamical
triangulations. The first article shows how to obtain a multicritical 2d model
by coupling the theory to hard dimers. The second explores the connection to
Ho\v{r}ava-Lifshitz gravity that is suggested by the time foliation and
establishes that in 2d the theories are equivalent. The last article does not
directly concern causal dynamical triangulations but Euclidian dynamical
triangulations with an additional measure term, which are examined to
understand whether they contain an extended phase without the need for a
preferred time foliation.
  Causal set theory uses an explicitly Lorentz invariant discretisation, which
introduces non-local effects.
  The second part of this thesis presents two articles in causal set theory.
The first explicitly calculates closed form expressions for the d'Alembertian
operator in any dimension, which can be implemented in computer simulations.
The second develops a ruler to examine the manifoldlikeness of small regions in
a causal set, and can be used to recover locality.
