In light of the progress in cosmological observations, in gravitational waves detection and in particle physics, there are exciting perspectives for discoveries at the interface between classical and quantum theories. To make the most of this variety of new data, it becomes increasingly important to understand and model how classical and quantum fields propagate and influence the spacetime geometry, and how quantum phenomena manifest themselves on the large scale. This imperative raises difficult mathematical questions which require a refined understanding of asymptotic structures, field propagation and spacetime dynamics, and of the relationships of quantum degrees of freedom with geometry. From the mathematical point of view it is now a particularly exciting time to address these interconnected problems because of the broad advances in partial differential equations and in field quantization: the former has resulted in a comprehensive array of methods to describe black hole physics and scattering phenomena on curved spacetimes, whereas the latter has provided crucial clarity into problematic concepts and formalisms, paving also new paths towards capturing quantum effects induced by gravity. The main objective of the IHP programme will be to take these developments to the next level and to create an environment for an unprecedented exchange of ideas between expert mathematicians and physicists. The focus will be on topics among the following, both from the mathematical and physical perspective: - Quantum Field Theory on curved spacetimes - Semi-classical and effective theories of gravity - Spectral action principles for gravity and beyond - Asymptotic analysis and scattering on curved spacetimes - Geometry of null surfaces and the black hole entropy problem