Inspired by possible connections between gravity and foundational question in
quantum theory, we consider an approach for the adaptation of objective
collapse models to a general relativistic context. We apply these ideas to a
list of open problems in cosmology and quantum gravity, such as the emergence
of seeds of cosmic structure, the black hole information issue, the problem of
time in quantum gravity and, in a more speculative manner, to the nature of
dark energy and the origin of the very special initial state of the universe.
We conclude that objective collapse models offer a rather promising path to
deal with all of these issues.