  We develop a superfield approach to compute chiral anomalies in general
${\cal N}=(1,0)$ supersymmetric gauge theories in six dimensions. Within the
harmonic-superspace formulation for these gauge theories, the anomalous
contributions to the effective action only come from matter and ghost
hypermultiplets. By studying the short-distance behaviour of the propagator for
the hypermultiplet coupled to a background vector multiplet, we compute the
covariant and consistent chiral anomalies. We also provide a superform
formulation for the non-abelian anomalous current multiplet in general ${\cal
N}=(1,0)$ supersymmetric gauge theories.
