  We present recent results on the enumeration of $q$-coloured planar maps,
where each monochromatic edge carries a weight $\nu$. This is equivalent to
weighting each map by its Tutte polynomial, or to solving the $q$-state Potts
model on random planar maps. The associated generating function, obtained by
Olivier Bernardi and the author, is differentially algebraic. That is, it
satisfies a (non-linear) differential equation. The starting point of this
result is a functional equation written by Tutte in 1971, which translates into
enumerative terms a simple recursive description of planar maps. The proof
follows and adapts Tutte's solution of properly $q$-coloured triangulations
(1973-1984).
  We put this work in perspective with the much better understood enumeration
of families of uncoloured planar maps, for which the recursive approach almost
systematically yields algebraic generating functions. In the past 15 years,
these algebraicity properties have been explained combinatorially by
illuminating bijections between maps and families of plane trees. We survey
both approaches, recursive and bijective.
  Comparing the coloured and uncoloured results raises the question of
designing bijections for coloured maps. No complete bijective solution exists
at the moment, but we present bijections for certain specialisations of the
general problem. We also show that for these specialisations, Tutte's
functional equation is much easier to solve that in the general case.
  We conclude with some open questions.
