The rum industry is currently worth USD 16 billion, with production concentrated in tropical countries of the Caribbean and Asia-Pacific regions. The primary feedstock for rum production is sugar cane molasses, a by-product of sugar refineries. The main variables known to affect rum quality include the composition of the molasses, the length of fermentation, and the type of barrels and length of time used for aging the rum. The goal of this review is to provide an overview of the impact of these variables on rum quality, and to highlight current challenges and opportunities in the production of rum from molasses. In order to achieve this, we review the relevant contemporary scientific literature on these topics. The major contemporary challenges in the rum production industry include minimising the effects of variability in feedstock quality, ensuring the fermentation process runs to completion, preventing microbial contamination, and the selection and maintenance of yeast strains providing optimum ethanol production. Stringent quality management practices are required to ensure consistency in the quality and organoleptic properties of the rum from batch to batch. Further research is required to fully understand the influences of many of these variables on the final quality of the rum produced.