This is an announcement for the paper "Saturating constructions for normed spaces" by Stanislaw J. Szarek and Nicole Tomczak-Jaegermann . Abstract: We prove several results of the following type: given finite dimensional normed space V there exists another space X with log(dim X) = O(log(dim V)) and such that every subspace (or quotient) of X, whose dimension is not "too small," contains a further subspace isometric to V. This sheds new light on the structure of such large subspaces or quotients (resp., large sections or projections of convex bodies) and allows to solve several problems stated in the 1980s by V. Milman. The proofs are probabilistic and depend on careful analysis of images of convex sets under Gaussian linear maps. Archive classification: Functional Analysis; Probability Mathematics Subject Classification: 46B20; 52A21; 52A22; 60D05 Remarks: 27 p., LATEX The source file(s), SzarekTomczakSat1.tex: 71711 bytes, is(are) stored in gzipped form as 0407233.gz with size 25kb. The corresponding postcript file has gzipped size 105kb. Submitted from: szarek@cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0407233 or http://arXiv.org/abs/math.FA/0407233 or by email in unzipped form by transmitting an empty message with subject line uget 0407233 or in gzipped form by using subject line get 0407233 to: math@arXiv.org.