||The kernel method is a powerful method which allows to solve functional (system of) equations which involve more unknowns than the number of given equations. Such functional equations are ubiquitous in combinatorics,
e.g. for the enumeration of lattice paths (the combinatorial discrete analogue of Brownian motion theory),
or planar maps, or other constrained combinatorial structures. Not only it gives some nice explicit formulas, but it is also the key to access to enumeration and asymptotics, and thus to typical properties of these structures: limit laws, phase transitions. The techniques used are typical of the philosophy of the wonderful book "Analytic Combinatorics" by Flajolet and Sedgewick.