The creation and perception of humour is a fundamental human trait, positioning its computational understanding as one of the most challenging tasks in natural language processing (NLP). As an abstract, creative, and frequently context-dependent construct, humour requires extensive reasoning to understand and create, making it a pertinent task for assessing the common-sense knowledge and reasoning abilities of modern large language models (LLMs). In this work, we survey the landscape of computational humour as it pertains to the generative tasks of creation and explanation. We observe that, despite the task of understanding humour bearing all the hallmarks of a foundational NLP task, work on generating and explaining humour beyond puns remains sparse, while state-of-the-art models continue to fall short of human capabilities. We bookend our literature survey by motivating the importance of computational humour processing as a subdiscipline of NLP and presenting an extensive discussion of future directions for research in the area that takes into account the subjective and ethically ambiguous nature of humour.