In this paper, we report on the observational performance of the Swift Ultra-violet/Optical Telescope (UVOT) in response to the Gravitational Wave alerts announced by the Advanced Laser Interferometer Gravitational Wave Observatory and the Advanced Virgo detector during the O3 period. We provide the observational strategy for follow-up of GW alerts and provide an overview of the processing and analysis of candidate optical/UV sources. For the O3 period, we also provide a statistical overview and report on serendipitous sources discovered by Swift/UVOT. Swift followed 18 gravitational-wave candidate alerts, with UVOT observing a total of 424 deg^2. We found 27 sources that changed in magnitude at the 3 sigma level compared with archival u or g-band catalogued values. Swift/UVOT also followed up a further 13 sources reported by other facilities during the O3 period. Using catalogue information, we divided these 40 sources into five initial classifications: 11 candidate active galactic nuclei (AGN)/quasars, 3 Cataclysmic Variables (CVs), 9 supernovae, 11 unidentified sources that had archival photometry and 6 uncatalogued sources for which no archival photometry was available. We have no strong evidence to identify any of these transients as counterparts to the GW events. The 17 unclassified sources are likely a mix of AGN and a class of fast-evolving transient, and one source may be a CV.
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