Enumeration of planar maps: from the quadratic method to quantum gravity

Among the many contributions of William Tutte is the enumerative theory 
of planar maps, which he laid down in his series of "Census" papers in 
1962-1963. Unexpectedly, his results had an echo in theoretical physics 
in the eighties, where the enumeration of maps was first connected with 
random matrix theory, and then applied successfully to the study of 
two-dimensional quantum gravity (maps being viewed as discrete random 
surfaces). Since the turn of this century, the subject further developed 
into the bijective theory of planar maps: in addition to giving 
elementary proofs of Tutte's results, this allowed to gain insight into 
the geometry of random planar maps, culminating in the definition of the 
Brownian map and related deep concepts in probability theory. I will 
attempt to present a (partial and personal) overview of this fascinating 
story.