  We give a sequence of equivalent formulations of the $ADE$ and $\hat A\hat
D\hat E$ height models defined on a random triangulated surface: random
surfaces immersed in Dynkin diagrams, chains of coupled random matrices,
Coulomb gases, and multicomponent Bose and Fermi systems representing soliton
$\tau$-functions. We also formulate a set of loop-space Feynman rules allowing
to calculate easily the partition function on a random surface with arbitrary
topology. The formalism allows to describe the critical phenomena on a random
surface in a unified fashion and gives a new meaning to the $ADE$
classification.
