  Einstein's field equation of General Relativity (GR) has been known for over
100 years, yet it remains challenging to solve analytically in strongly
relativistic regimes, particularly where there is a lack of a priori symmetry.
Numerical Relativity (NR) - the evolution of the Einstein Equations using a
computer - is now a relatively mature tool which enables such cases to be
explored. In this thesis, a description is given of the development and
application of a new Numerical Relativity code, GRChombo. GRChombo uses the
standard BSSN formalism, incorporating full adaptive mesh refinement (AMR) and
massive parallelism via the Message Passing Interface (MPI). The AMR capability
permits the study of physics which has previously been computationally
infeasible in a full 3+1 setting. The functionality of the code is described,
its performance characteristics are demonstrated, and it is shown that it can
stably and accurately evolve standard spacetimes such as black hole mergers. We
use GRChombo to study the effects of inhomogeneous initial conditions on the
robustness of small and large field inflationary models. and investigate the
critical behaviour which occurs in the collapse of both spherically symmetric
and asymmetric scalar field bubbles.

General Relativity and Quantum Cosmology