  In the 1970s, Tutte developed a clever algebraic approach, based on certain
"invariants" , to solve a functional equation that arises in the enumeration of
properly colored triangulations. The enumeration of plane lattice walks
confined to the first quadrant is governed by similar equations, and has led in
the past decade to a rich collection of attractive results dealing with the
nature (algebraic, D-finite or not) of the associated generating function,
depending on the set of allowed steps. We first adapt Tutte's approach to prove
(or reprove) the algebraicity of all quadrant models known or conjectured to be
algebraic (with one small exception). This includes Gessel's famous model, and
the first proof ever found for one model with weighted steps. To be applicable,
the method requires the existence of two rational functions called invariant
and decoupling function respectively. When they exist, algebraicity comes out
(almost) automatically. Then, we move to an analytic viewpoint which has
already proved very powerful, leading in particular to integral expressions of
the generating function in the non-D-finite cases, as well as to proofs of
non-D-finiteness. We develop in this context a weaker notion of invariant. Now
all quadrant models have invariants, and for those that have in addition a
decoupling function, we obtain integral-free expressions of the generating
function, and a proof that this series is differentially algebraic (that is,
satisfies a non-linear differential equation).
