Within causal set theory, growth dynamics form a novel realisation of the path integral for quantum gravity. This thesis presents recent developments in growth dynamics, focusing on general covariance and quantization.
In causal set theory, where spacetime takes the form of a discrete causal set, general covariance takes the form of label-invariance. Here we present the first manifestly covariant growth dynamics, namely models of random unlabeled graphs. These models, like their label-dependent predecessors, are classically stochastic. The decoherence functional offers a stochastic-like formulation of quantum theory particularly suited to quantum gravity and a proposal for a decoherence functional for causal sets based on growth dynamics has previously been put forward, but it was shown to fail in certain cases due to a technical pathology. We provide new criteria for when the procedure is well-defined and apply these to obtain the first known examples of quantum dynamics for causal sets. We generalise the construction of the decoherence functional to a wide class of growth dynamics and discuss its application to dynamics which give rise to bouncing cosmologies. Finally, we explore whether growth dynamics, labeled and unlabeled, can accommodate cosmologies in which time has no beginning.