  Conserved and commuting charges are investigated in both bosonic and
supersymmetric classical chiral models, with and without Wess-Zumino terms. In
the bosonic theories, there are conserved currents based on symmetric invariant
tensors of the underlying algebra, and the construction of infinitely many
commuting charges, with spins equal to the exponents of the algebra modulo its
Coxeter number, can be carried out irrespective of the coefficient of the
Wess-Zumino term. In the supersymmetric models, a different pattern of
conserved quantities emerges, based on antisymmetric invariant tensors. The
current algebra is much more complicated than in the bosonic case, and it is
analysed in some detail. Two families of commuting charges can be constructed,
each with finitely many members whose spins are exactly the exponents of the
algebra (with no repetition modulo the Coxeter number). The conserved
quantities in the bosonic and supersymmetric theories are only indirectly
related, except for the special case of the WZW model and its supersymmetric
extension.
