  I hesitated for a long time before giving shape to these notes, originally
intended for preliminary reading by the attendees to the Summer School "New
paths towards quantum gravity" (Holbaek Bay, Denmark, May 2008). At the end, I
decide against just selling my mathematical wares, and for a survey,
necessarily very selective, but taking a global phenomenological approach to
its subject matter. After all, non-commutative geometry does not purport yet to
solve the riddle of quantum gravity; it is more of an insurance policy against
the probable failure of the other approaches. The plan is as follows: the
introduction invites students to the fruitful doubts and conundrums besetting
the application of even classical gravity. Next, the first experiments
detecting quantum gravitational states inoculate us a healthy dose of
skepticism on some of the current ideologies. In Section 3 we look at the
action for general relativity as a consequence of gauge theory for quantum
tensor fields. Section 4 briefly deals with the unimodular variants. Section 5
arrives at non-commutative geometry. I am convinced that, if this is to play a
role in quantum gravity, commutative and non-commutative manifolds must be
treated on the same footing; which justifies the place granted to the
reconstruction theorem. Together with Section 3, this part constitutes the main
body of the notes. Only very summarily at the end of this section we point to
some approaches to gravity within the non-commutative realm. The last section
delivers a last dose of skepticism. My efforts will have been rewarded if
someone from the young generation learns to mistrust current mindsets.
