  In the light of the recent discovery of the $\Xi^{++}_{cc}$ by LHCb
collaboration, we study the stable doubly heavy tetraquarks. These states are
compact exotic hadrons which can be approximated as the diquark-anti-diquark
correlations. In the flavor SU(3) symmetry, they form an SU(3) triplet or
anti-sextet. The spectra of the stable doubly heavy tetraquark states are
predicted by Sakharov-Zeldovich formula. We find that the
$T^+_{cc\bar{u}\bar{d}}({\bf 3})$ is about 16MeV below the $DD^*$ threshold,
while $T^{-}_{bb\bar{u}\bar{d}}({\bf 3})$ is about 73MeV below the $BB^*$
threshold. We then study the semileptonic and nonleptonic weak decays of the
stable doubly heavy tetraquark states. The decay amplitudes are parametrized in
terms of SU(3)-irreducible amplitudes. Ratios between decay widths of different
channels are also derived. At the end, we collect the Cabibbo allowed two-body
and three-body decay channels, which are most promising to search for the
stable doubly heavy tetraquark states at LHCb and Belle II experiments.
