  This paper is mainly a review of the multi--Hamiltonian nature of Toda and
generalized Toda lattices corresponding to the classical simple Lie groups but
it includes also some new results. The areas investigated include master
symmetries, recursion operators, higher Poisson brackets, invariants and group
symmetries for the systems. In addition to the positive hierarchy we also
consider the negative hierarchy which is crucial in establishing the
bi--Hamiltonian structure for each particular simple Lie group. Finally, we
include some results on point and Noether symmetries and an interesting
connection with the exponents of simple Lie groups. The case of exceptional
simple Lie groups is still an open problem.
