To supplement the already known classification of traces on classical pseudodifferential
operators, we present a classification of traces on the algebras of odd-class
pseudodifferential operators of non-positive order acting on smooth functions on a closed
odd-dimensional manifold. By means of the one to one correspondence between continuous
traces on Lie algebras and determinants on the associated regular Lie groups, we give
a classification of determinants on the group associated to the algebra of odd-class pseudodifferential
operators with fixed non-positive order. At the end we discuss two possible ways
to extend the definition of a determinant outside a neighborhood of the identity on the Lie
group associated to the algebra of odd-class pseudodifferential operators of order zero.