The role extragalactic class I methanol masers play in their host galaxies is currently not well understood. With only a few known examples of these masers to date, investigating their nature is very difficult. However, determining the physical conditions and environments responsible for these masers will provide a useful tool for the study of external galaxies. In this thesis we present over a hundred hours of Australia Telescope Compact Array (ATCA) observations aimed at expanding our available sample size, increasing our understanding of the regions in which these masers form, and making comparison with class I methanol emission towards Galactic regions. We present the first detection of 36.2-GHz class I methanol maser emission toward NGC 4945, which is the second confirmed detection of this masing transition and the brightest class I maser region ever detected. This emission is offset from the nucleus of the galaxy and is likely associated with the interface region between the bar and south-eastern spiral arm. Additionally, we report the first detection of the 84.5-GHz class I maser line toward NGC 253, making NGC 253 the only external galaxy displaying 3 separate masing transitions of methanol. Upper limits on the 36.2- class I and 37.7-GHz class II maser lines are also reported for 12 other galaxies that were searched as part of this project. Alongside observations of the class I methanol lines, we also observed various thermal and masing molecular lines towards these sources. This includes the imaging of 7-mm HC\\(_3\\)N, CS and SiO lines towards NGC 4945, which had not previously been observed at high resolution before, along with some 3-mm lines, such as HNCO, towards NGC 253. Long term monitoring of the 36.2-GHz transition from NGC 4945 and NGC 253 was also performed. This monitoring produced detections of new maser regions in both NGC 4945 and NGC 253 and provided the first evidence for variability from these extragalactic class I masers, with the luminosity of the new maser region in NGC 4945 increasing by approximately 300 percent compared to past observations. We determine this is likely the result of excitation changes in the maser environments, however, further monitoring is required to accurately quantify this variability. We also present interferometric imaging of the 95.2-GHz class I methanol masers towards 32 6.7-GHz class II maser sources along with detailed comparison between the class I (36.2-, 44.1- and 95.2-GHz) and class II (6.7- and 12.2-GHz) lines in these high-mass star formation regions. Flux-density ratios for matched (spatially associated) 44.1- and 95.2-GHz class I masers shows that the 3:1 ratio previously reported in the literature holds true even at parsec scales. Investigation of our statistically complete sample indicates that 95.2-GHz class I methanol masers do not preferentially favour more evolved sources. Our observations reinforce that when using class I methanol masers to investigate shock fronts and star-formation region dynamics, observation of multiple class I transitions is necessary in order to form a complete picture.
Copyright 2020 the author Chapter 2 is the version of record of an article accepted for publication in Monthly Notices of the Royal Astronomical Society Copyright: 2018 the authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved Chapter 4 is the version of record of an article accepted for publication in Monthly Notices of the Royal Astronomical Society Copyright: 2018 the authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. All rights reserved Chapter 6 is a article has been accepted for publication in Monthly Notices of the Royal Astronomical Society. Published by Oxford University Press on behalf of the Royal Astronomical Society