  The lifetime of the remnant produced by the merger of two neutron stars can
provide a wealth of information on the equation of state of nuclear matter and
on the processes leading to the electromagnetic counterpart. Hence, it is
essential to determine when this lifetime is the shortest, corresponding to
when the remnant has a mass equal to the threshold mass, $M_{\rm th}$, to
prompt collapse to a black hole. We report on the results of more than 360
simulations of merging neutron-star binaries covering 40 different
configurations differing in mass ratio and spin of the primary. Using this
data, we have derived a quasi-universal relation for $M_{\rm th}$ and expressed
its dependence on the mass ratio and spin of the binary. The new expression
recovers the results of Koeppel et al for equal-mass, irrotational binaries and
reveals that $M_{\rm th}$ can increase (decrease) by $5\%~(10\%)$ for binaries
that have spins aligned (anti-aligned) with the orbital angular momentum and
provides evidence for a non-monotonic dependence of $M_{\rm
  th}$ on the mass asymmetry in the system. Finally, we extend to
unequal-masses and spinning binaries the lower limits that can be set on the
stellar radii once a neutron-star binary is detected, illustrating how the
merger of an unequal-mass, rapidly spinning binary can significantly constrain
the allowed values of the stellar radii.
