  We characterize those classes $\ccc$ of separable Banach spaces admitting a
separable universal space $Y$ (that is, a space $Y$ containing, up to
isomorphism, all members of $\ccc$) which is not universal for all separable
Banach spaces. The characterization is a byproduct of the fact, proved in the
paper, that the class $\mathrm{NU}$ of non-universal separable Banach spaces is
strongly bounded. This settles in the affirmative the main conjecture form
\cite{AD}. Our approach is based, among others, on a construction of
$\llll_\infty$-spaces, due to J. Bourgain and G. Pisier. As a consequence we
show that there exists a family $\{Y_\xi:\xi<\omega_1\}$ of separable,
non-universal, $\llll_\infty$-spaces which uniformly exhausts all separable
Banach spaces. A number of other natural classes of separable Banach spaces are
shown to be strongly bounded as well.
