  A pedagogical presentation of integrable models with special reference to the
Toda lattice hierarchy has been attempted. The example of the KdV equation has
been studied in detail, beginning with the infinite conserved quantities and
going on to the Lax formalism for the same. We then go on to symplectic
manifolds for which we construct the Lax operator. This formalism is applied to
Toda Lattice systems. The Zakharov Shabat formalism aimed at encompassing all
integrable models is also covered after which the zero curvature condition and
its fallout are discussed. We then take up Toda Field Theories and their
connection to W algebras via the Hamiltonian reduction of the WZNW model.
Finally, we dwell on the connection between four dimensional Yang Mills
theories and the KdV equation along with a generalization to supersymmetry.
