  The Causal Set approach to quantum gravity asserts that spacetime, at its
smallest length scale, has a discrete structure. This discrete structure takes
the form of a locally finite order relation, where the order, corresponding
with the macroscopic notion of spacetime causality, is taken to be a
fundamental aspect of nature.
  After an introduction to the Causal Set approach, this thesis considers a
simple toy dynamics for causal sets. Numerical simulations of the model provide
evidence for the existence of a continuum limit. While studying this toy
dynamics, a picture arises of how the dynamics can be generalized in such a way
that the theory could hope to produce more physically realistic causal sets. By
thinking in terms of a stochastic growth process, and positing some fundamental
principles, we are led almost uniquely to a family of dynamical laws
(stochastic processes) parameterized by a countable sequence of coupling
constants. This result is quite promising in that we now know how to speak of
dynamics for a theory with discrete time. In addition, these dynamics can be
expressed in terms of state models of Ising spins living on the relations of
the causal set, which indicates how non-gravitational matter may arise from the
theory without having to be built in at the fundamental level.
  These results are encouraging in that there exists a natural way to transform
this classical theory, which is expressed in terms of a probability measure, to
a quantum theory, expressed in terms of a quantum measure. A sketch as to how
one might proceed in doing this is provided. Thus there is good reason to
expect that Causal Sets are close to providing a background independent theory
of quantum gravity.
