  Non-conformal supercurrents in six dimensions are described, which contain
the trace of the energy-momentum tensor and the gamma-trace of the
supersymmetry current amongst their component fields. Within the superconformal
approach to ${\cal N} = (1, 0)$ supergravity, we present various distinct
non-conformal supercurrents, one of which is associated with an ${\cal O}(2)$
(or linear) multiplet compensator, while another with a tensor multiplet
compensator. We also derive an infinite class of non-conformal supercurrents
involving ${\cal O}(n)$ multiplets with $n > 2$. As an illustrative example we
construct the relaxed hypermultiplet in supergravity. Finally, we put forward a
non-conformal supercurrent in the ${\cal N} = (2, 0)$ supersymmetric case.
