On classes of Banach spaces admitting "small" universal spaces

We characterize those classes $\text{ccc}$ of separable Banach spaces admitting a separable universal space $Y$ (that is, a space $Y$ containing, up to isomorphism, all members of $\text{ccc}$) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class $\mathrm{N}\mathrm{U}$ 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 ${\text{llll}}_{\mathrm{\infty }}$-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, ${\text{llll}}_{\mathrm{\infty }}$-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.