Diffuse Extragalactic Emission at Low Frequencies within the Epoch of Reionization 0-hour Field

Low-frequency radio imaging of the southern sky has become available with the advent of the Murchison Widefield Array (MWA). The topic of this thesis is the study of extended, low-frequency radio emission, with a primary focus on the non-thermal synchrotron emission associated with the intra-cluster medium (ICM) of galaxy clusters. We do not limit the study to such emission, however, and investigate a small sample of other interesting and extended radio emission from objects in the southern sky. A significant portion of this work is invested in detecting, and characterising, extended, diffuse radio emission from galaxy clusters within a 45 degree by 45 degree region of the southern sky centred on R.A. = 0 hours, decl. = -27 degrees. This field is chosen as a deep MWA image has been made available which is sensitive to extended structures. Within the field we search for low-frequency, diffuse cluster emission, previously detected or otherwise. In doing so we find 34 diffuse radio sources, 3 of which are newly detected haloes, 1 newly detected relic with many new candidates of each. Further, we detect a new phoenix candidate as well as 2 candidate dead radio galaxies at the centre of clusters. We confirm previous observations of such emission as well, and measure properties such as their integrated flux densities, spectral indices, and sizes where possible. We compare our sample of haloes with previously detected haloes and revisit established scaling relations of the radio halo power with the cluster X-ray luminosity and mass. We find that both scaling relations are consistent with previous findings despite the increase in sample size, though note that the raw scatter in the data for best-fitting parameters increases with increase in sample size. In this, we demonstrate the utility of low-frequency radio telescopes like the MWA in detecting such emission, showing that the MWA is pushing into higher-redshift, lower-mass systems, though we caution that the low resolution of the MWA can work against us. We follow-up on two galaxy clusters found to host extended emission - Abell S1136 and Abell S1063. In the case of Abell S1136 we observe the emission at its centre with the Australia Telescope Compact Array (ATCA) and determine the presence of a core, suggesting the emission to be that of an ancient episode of an active galactic nucleus in the central elliptical of the cluster, ESO 470-G020. After reducing archival ATCA data for Abell S1063 we find no evidence of a halo and consider the source to be constructed of blended point sources. We close with a description of a strong double-lobed radio source associated with a non-elliptical host ESO 472-G013, likely a spiral or irregular galaxy, that was found serendipitously whilst searching for diffuse cluster emission. We explore the host within the context of star-formation, and consider the possible origins of the AGN and lobes due to interaction with either the nearby spiral, ESO 472-G012, or a past or ongoing merger event