This is an announcement for the paper "Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive" by Stanislaw J. Szarek, Elisabeth Werner and Karol Zyczkowski. Abstract: We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiolkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensional probability and geometry of Banach spaces, some of which are not standard. Archive classification: quant-ph math.FA Remarks: 34 pages in Latex including 3 figures in eps The source file(s), , is(are) stored in gzipped form as with size . The corresponding postcript file has gzipped size . Submitted from: karol@tatry.if.uj.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0710.1571 or http://arXiv.org/abs/0710.1571 or by email in unzipped form by transmitting an empty message with subject line uget 0710.1571 or in gzipped form by using subject line get 0710.1571 to: math@arXiv.org.