  We analyze multi--matrix chain models. They can be considered as
multi--component Toda lattice hierarchies subject to suitable coupling
conditions. The extension of such models to include extra discrete states
requires a weak form of integrability. The discrete states of the $q$--matrix
model are organized in representations of $sl_q$. We solve exactly the
Gaussian--type models, of which we compute several all-genus correlators. Among
the latter models one can classify also the discretized $c=1$ string theory,
which we revisit using Toda lattice hierarchy methods. Finally we analyze the
topological field theory content of the $2q$--matrix models: we define primary
fields (which are $\infty^q$), metrics and structure constants and prove that
they satisfy the axioms of topological field theories. We outline a possible
method to extract interesting topological field theories with a finite number
of primaries.
