  We show that continuous bounded group cohomology stabilizes along the
sequences of real or complex symplectic Lie groups, and deduce that bounded
group cohomology stabilizes along sequences of lattices in them, such as
$(\mathrm{Sp}_{2r}(\mathbb{Z}))_{r \geq 1}$ or
$(\mathrm{Sp}_{2r}(\mathbb{Z}[i]))_{r \geq 1}$. Our method is based on a
general stability criterion which extends Quillen's method to the functional
analytic setting of bounded cohomology. This criterion is then applied to a new
family of complexes associated to symplectic polar spaces, which we call
symplectic Stiefel complexes; similar complexes can also be defined for other
families of classical groups acting on polar spaces.
