  We theoretically propose a nonreciprocal spin Seebeck effect, i.e.,
nonreciprocal spin transport generated by a temperature gradient, in
antiferromagnetic insulators with broken inversion symmetry. We find that
nonreciprocity in antiferromagnets has rich properties not expected in
ferromagnets. In particular, we show that polar antiferromagnets, in which the
crystal lacks the spatial inversion symmetry, exhibit perfect nonreciprocity
--- one-way spin current flow irrespective of the direction of the temperature
gradient. We also show that nonpolar centrosymmetric crystals can exhibit
nonreciprocity when a magnetic order breaks the inversion symmetry, and in this
case, the direction of the nonreciprocal flow can be controlled by reversing
the magnetic domain. As their representatives, we calculate the nonreciprocal
spin Seebeck voltages for the polar antiferromagnet $\alpha$-Cu$_2$V$_2$O$_7$
and the honeycomb antiferromagnet MnPS$_3$, while varying temperature and
magnetic field.
