Urgent and emergency healthcare requires accurate demand and capacity modelling to ensure the rapid deployment of medical resources and patients quickly assigned to the correct healthcare pathway. A suite of forecasting, queueing, simulation and optimisation models have been developed for capturing time-dependent and time-critical scenarios that have been applied across multiple NHS settings. For example, our research has helped ambulance services to better forecast demand, optimise the geographical allocation of emergency vehicles and roster staff. Our mathematical modelling work was also key to the Welsh Government’s investment in the Single Cancer Pathway, which improved nationwide urgent cancer treatment, as well as informing the staff profile required to launch the Welsh NHS 111 service.