  In this paper we describe in some detail the representation of the
topological $CP^1$ model in terms of a matrix integral which we have introduced
in a previous article. We first discuss the integrable structure of the $CP^1$
model and show that it is governed by an extension of the 1-dimensional Toda
hierarchy. We then introduce a matrix model which reproduces the sum over
holomorphic maps from arbitrary Riemann surfaces onto $CP^1$. We compute
intersection numbers on the moduli space of curves using geometrical method and
show that the results agree with those predicted by the matrix model. We also
develop a Landau-Ginzburg (LG) description of the $CP^1$ model using a
superpotential $e^X+e^{t_{0,Q}}e^{-X}$ given by the Lax operator of the Toda
hierarchy ($X$ is the LG field and $t_{0,Q}$ is the coupling constant of the
K\"ahler class). The form of the superpotential indicates the close connection
between $CP^1$ and $N=2$ supersymmetric sine-Gordon theory which was noted some
time ago by several authors. We also discuss possible generalizations of our
construction to other manifolds and present a LG formulation of the topological
$CP^2$ model.
