Despite the apparently unrealistic assumption of infinite resources, infinite-server queueing models have played a central role in the development of queueing theory and its applications. Healthcare modelling applications have certainly benefited from these models, where arguably their greatest importance has been to provide the basis for the analysis of "offered load" in systems with single or multiple nodes with multiple servers and timevarying arrivals. In this paper, we provide a review of major healthcare applications to date, identifying and consolidating the underpinning theoretical results and commenting on the nature of the applications. We conclude by identifying potential further healthcare applications, their relationships to existing theory and methods, and the need for new theory and methods, including the use of infinite-server models alongside other modelling methodologies.