  We present an exact solution of the $O(n)$ model on a random lattice. The
coupling constant space of our model is parametrized in terms of a set of
moment variables and the same type of universality with respect to the
potential as observed for the one-matrix model is found. In addition we find a
large degree of universality with respect to $n$; namely for $n\in ]-2,2[$ the
solution can be presented in a form which is valid not only for any potential,
but also for any $n$ (not necessarily rational). The cases $n=\pm 2$ are
treated separately. We give explicit expressions for the genus zero
contribution to the one- and two-loop correlators as well as for the genus one
contribution to the one-loop correlator and the free energy. It is shown how
one can obtain from these results any multi-loop correlator and the free energy
to any genus and the structure of the higher genera contributions is described.
Furthermore we describe how the calculation of the higher genera contributions
can be pursued in the scaling limit.
