  The canonical approach to quantum gravity has been put on a firm mathematical
foundation in the recent decades. Even the quantum dynamics can be rigorously
defined, however, due to the tremendously non-polynomial character of the
gravitational interaction, the corresponding Wheeler-DeWitt operator valued
distribution suffers from quantisation ambiguities that need to be fixed.
  In a very recent series of works we have employed methods from constructive
quantum field theory in order to address those ambiguities. Constructive QFT
trades quantum fields for random variables and measures thereby phrasing the
theory in the language of quantum statistical physics. The connection to the
canonical formulation is made via Osterwalder-Schrader reconstruction. It is
well known in quantum statistics that the corresponding ambiguities in measures
can be fixed using renormalisation. The associated renormalisation flow can
thus be used to define a canonical renormalisation programme.
  The purpose of this article is to review and further develop these ideas and
to put them into context with closely related earlier and parallel programmes.
