  We calculate the nature of magnetic interactions in transition-metal doped
ZnO using the local spin density approximation and LSDA+\textit{U} method of
density functional theory. We investigate the following four cases: (i) single
transition metal ion types (Cr, Mn, Fe, Co, Ni and Cu) substituted at Zn sites,
(ii) substitutional magnetic transition metal ions combined with additional Cu
and Li dopants, (iii) substitutional magnetic transition metal ions combined
with oxygen vacancies and (iv) pairs of magnetic ion types (Co and Fe, Co and
Mn, etc.). Extensive convergence tests indicate that the calculated magnetic
ground state is unusually sensitive to the k-point mesh and energy cut-off, the
details of the geometry optimizations and the choice of the
exchange-correlation functional. We find that ferromagnetic coupling is
sometimes favorable for single type substitutional transition metal ions within
the local spin density approximation. However, the nature of magnetic
interactions changes when correlations on the transition-metal ion are treated
within the more realistic LSDA + \textit{U} method, often disfavoring the
ferromagnetic state. The magnetic configuration is sensitive to the detailed
arrangement of the ions and the amount of lattice relaxation, except in the
case of oxygen vacancies when an antiferromagnetic state is always favored.
