  We give a complete characterization of limiting interpolation spa\-ces for
the real method of interpolation using extrapolation theory. For this purpose
the usual tools (e.g., Boyd indices or the boundedness of Hardy type operators)
are not appropriate. Instead, our characterization hinges upon the boundedness
of some simple operators (e.g. $f\mapsto f(t^{2})/t$, or $f\mapsto f(t^{1/2}%
)$) acting on the underlying lattices that are used to control the $K$- and
$J$-functionals. Reiteration formulae, extending Holmstedt's classical
reiteration theorem to limiting spaces, are also proved and characterized in
this fashion. The resulting theory gives a unified roof to a large body of
literature that, using ad-hoc methods, had covered only special cases of the
results obtained here. Applications to Matsaev ideals, Grand Lebesgue spaces,
Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova limits, as well as a new vector
valued extrapolation theorems, are provided.
