This is an announcement for the paper "Partial Unconditionality" by S. J. Dilworth, E. Odell, Th. Schlumprecht and A. Zsak. Abstract: J. Elton proved that every normalized weakly null sequence in a Banach space admits a subsequence that is nearly unconditional which is a weak form of unconditionality. The notion of near-unconditionality is quantified by a constant $K(\delta)$ depending on a parameter $\delta \in (0,1]$. It is unknown if $\sup_{\delta>0} K(\delta) < \infty$. This problem turns out to be closely related to the question whether every infinite-dimensional Banach space contains a quasi-greedy basic sequence. The notion of a quasi-greedy basic sequence was introduced recently by S. V. Konyagin and V. N. Temlyakov. We present an extension of Elton's result which includes Schreier unconditionality. The proof involves a basic framework which we show can be also employed to prove other partial unconditionality results including that of convex unconditionality due to Argyros, Mercourakis and Tsarpalias. Various constants of partial unconditionality are defined and we investigate the relationships between them. We also explore the combinatorial problem underlying the $\sup_{\delta>0} K(\delta) < \infty$ problem and show that $\sup_{\delta>0} K(\delta) > 5/4$. Archive classification: Functional Analysis Mathematics Subject Classification: 46B15 Remarks: 50 pages The source file(s), partial_unconditionality.tex: 175575 bytes, is(are) stored in gzipped form as 0510609.gz with size 46kb. The corresponding postcript file has gzipped size 196kb. Submitted from: a.zsak@dpmms.cam.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0510609 or http://arXiv.org/abs/math.FA/0510609 or by email in unzipped form by transmitting an empty message with subject line uget 0510609 or in gzipped form by using subject line get 0510609 to: math@arXiv.org.