This is an announcement for the paper "On classes of Banach spaces admitting ``small" universal spaces" by Pandelis Dodos. Abstract: 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. Archive classification: math.FA math.LO Mathematics Subject Classification: 03E15, 46B03, 46B07, 46B15 Remarks: 25 pages, no figures The source file(s), Universal-ArXiv.tex: 81806 bytes, is(are) stored in gzipped form as 0805.2043.gz with size 24kb. The corresponding postcript file has gzipped size 143kb. Submitted from: pdodos@math.ntua.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0805.2043 or http://arXiv.org/abs/0805.2043 or by email in unzipped form by transmitting an empty message with subject line uget 0805.2043 or in gzipped form by using subject line get 0805.2043 to: math@arXiv.org.