This project brings together two important developments of recent years: the theory of Liouville quantum gravity and the theory of metric measure spaces. The objectives to be addressed are random geometries, their scaling and renormalization, and the role of curvature for the proper understanding of limiting complex structures. The focus will be on phenomena which depend on critical scaling. We will investigate the following aspects: the geometry of Liouville quantum gravity as random metric measure space, the Polyakov–Liouville measure and dynamics of metric measure spaces, the latter in particular with respect to critical singularities, homogenisation, and time-dependence.