What is quantum geometry? This question is becoming a popular leitmotiv in
theoretical physics and in mathematics. Conformal field theory may catch a
glimpse of the right answer. We review global aspects of the geometry of
conformal fields, such as duality and mirror symmetry, and interpret them
within Connes' non-commutative geometry. Extended version of lectures given by
the 2nd author at the Mathematical Quantum Theory Conference, Vancouver,
Canada, August 4 to 8, 1993Comment: 44 pages, latex file, 5 references adde