Modern atomic physics is increasingly dependent on the results of high-precision calculations to guide experiments and applications, especially in complex atoms with dense spectra. Many cutting-edge applications and experiments use atoms with open-shell electronic structure and strong configuration mixing, which require extremely large numerical bases to treat with any level of accuracy. At the same time, supercomputing clusters have seen huge increases in computational power, driven by increasingly large-scale parallelism across many distributed compute nodes. Consequently, modern atomic structure code must be designed to fully utilise massively-parallel computing resources if they are to keep up with the increasing demands of experimental studies.In this thesis, I present the results of work to modernise the AMBiT atomic structure software, which implements the configuration interaction with many-body perturbation theory (CI+MBPT) method, to take advantage of modern supercomputers. I present a detailed outline of the software engineering processes in converting AMBiT from an MPI-only model of parallelism to a hybrid MPI+OpenMP model, as well as the performance gains resulting from doing so. I show that the increased parallelism allows us to explore numerical saturation of the CI+MBPT method in open-shell atoms for the first time ever --- an investigation would not have been possible without the increased performance capabilities of modern supercomputers.I have applied the new AMBiT to calculations of atomic systems with a variety of electronic structures. Calculations of the highly-charged ions Sn7+-Sn10+, which are of experimental interest for their applications in extreme ultraviolet photolithography for semiconductor fabrication, show that AMBiT is highly efficient for ions with open d-shells. We achieve very close agreement with experimental spectra: CI+MBPT calculations differ from experiments by an average error of less than 1%. Additionally, calculations for two- and three-valent Lr+ and Lr demonstrate that CI+MBPT implemented in AMBiT can accurately calculate the spectra of superheavy elements - systems in which relativistic and QED effects are significant. My calculations for Lr and Lr+ also show that our CI+MBPT implementation is competitive with other cutting-edge methods for relativistic atomic structure calculations.This research shows that the accuracy of the CI+MBPT method, when scaled out to numerical saturation, is competitive with best-in-class methods for atomic structure calculations and can continue to serve as a workhorse for next-generation atomic structure calculations. Following its release, AMBiT has been used by multiple research groups for calculations of highly-charged ion clocks and precision tests for physics beyond the standard model. Furthermore, this research should put AMBiT and CI+MBPT in a strong position to scale up and capitalise on future gains in high-performance computing technology.