  We study tetraquarks in large $N$ QCD with heavy quarks, in the domain where
non-relativistic quantum mechanics offers an adequate approximation. Within the
regime of validity of the Born-Oppenheimer approximation, we systematically
study and explicitly construct tetraquark states. At leading order in the $1/N$
expansion, the bound spectrum consists of free mesons, while the $1/N$
corrections give rise to a Born-Oppenheimer potential that can bind the mesons
into tetraquarks. We find two different types of tetraquarks, each endowed with
distinct color-spatial wavefunctions. These states arise in the presence of an
$\mathcal{O}(N)$ mass hierarchy between the quarks and the antiquarks. We
provide a quantitative argument indicating that only for such a hierarchy is
the ground state of the system a tetraquark. We discuss what the extrapolation
of our results to realistic values of the parameters may imply for the QCD
tetraquark states.
