  We solve the loop equations to all orders in $1/N^2$, for the Chain of
Matrices matrix model (with possibly an external field coupled to the last
matrix of the chain). We show that the topological expansion of the free
energy, is, like for the 1 and 2-matrix model, given by the symplectic
invariants of the associated spectral curve. As a consequence, we find the
double scaling limit explicitly, and we discuss modular properties, large $N$
asymptotics. We also briefly discuss the limit of an infinite chain of matrices
(matrix quantum mechanics).
