Results on the finite nonperiodic Toda lattice are extended to some ge
neralizations of the system: The relativistic Toda lattice, the genera
lized Toda lattice associated with simple Lie groups and the full Kost
ant-Toda lattice. The areas investigated include master symmetries, re
cursion operators, higher Poisson brackets, invariants, and group symm
etries for the systems. A survey of previous work on the classical Tod
a lattice is also included.