  In this paper, a systematic investigation is carried out for the general
solvability of multi-dimensional backward stochastic Volterra integral
equations (BSVIEs) with the generators being super-linear in the adjustment
variable $Z$. Two major situations are discussed: (i) When the free term is
bounded with the dependence of the generator on $Z$ being of ``diagonally
strictly'' quadratic growth and being sub-quadratically coupled with
off-diagonal components; (ii) When the free term is unbounded having
exponential moments of arbitrary order with the dependence of the generator on
$Z$ being diagonally no more than quadratic and being independent of
off-diagonal components. Besides, for the case that the generator is
super-quadratic in $Z$, some negative results are presented.
