Innovative problem solving is critical for all spheres of organised endeavour, including science and industry, and thus forms the cornerstone of a successful society. Such creative thinking often re-quires suppression of preconceptions and restructuring of existing knowledge. Pioneering work has shown that sleep facilitates problem solving, but exactly how, and which sleep characteristics are important, remain to be determined. We know that recent experiences are replayed in sleep, and that in Slow Wave Sleep (SWS) this replay integrates new knowledge with old. The role of such replay in Rapid Eye Movement (REM) sleep, a stage which is strongly linked to creativity, is unknown. I have proposed a model which combines physiology, behavioural studies, and computational modelling to make testable predictions about the complimentary contributions of memory replay in REM and SWS to problem solving. SolutionSleep has tested this model through explicit manipulation of memory replay in sleep using a recently developed technique to trigger memory replay and a pioneering method for quantifying triggered replay. This work has studied memory replays in both REM and SWS in detail and described many of their characteristics, for instance finding that in SWS such replays are temporally compressed and tend to occur at the peaks of slow oscillations. We have also shown that triggered memory replay in SWS can facilitate both gist abstraction and the formation of seemingly difficult connections between distant concepts, while triggering replay in REM can help emotionally arousing information to become less upsetting. Our computational model of memory replay in sleep has revealed that there are specific windows of opportunity when triggering replay is more likely to be effective. Immediate consequences of this work include a translational project harnessing memory replay to develop treatments for Depression and PTSD. Longer range consequences will include a project studying how replay can be harnessed to facilitate problem solving. Overall, the findings of SolutionSleep have revolutionised the understanding of how sleep impacts upon some of our most important cognitive abilities—memory and problem solving, and how these interact with mental health.