This is an announcement for the paper "Greedy bases for Besov spaces" by S. J. Dilworth, D. Freeman, E. Odell and Th. Schlumprecht. Abstract: We prove thatthe Banach space $(\oplus_{n=1}^\infty \ell_p^n)_{\ell_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1\leq p \leq\infty$ and $1<q<\infty$. Furthermore, the Banach spaces $(\oplus_{n=1}^\infty \ell_p^n)_{\ell_1}$, with $1<p\le \infty$, and $(\oplus_{n=1}^\infty \ell_p^n)_{c_0}$, with $1\le p<\infty$ do not have a greedy bases. We prove as well that the space $(\oplus_{n=1}^\infty \ell_p^n)_{\ell_q}$ has a 1-greedy basis if and only if $1\leq p=q\le \infty$. Archive classification: math.FA Mathematics Subject Classification: 46B15, 41A65 The source file(s), dfos_greedy_101609.tex: 45739 bytes, is(are) stored in gzipped form as 0910.3867.gz with size 14kb. The corresponding postcript file has gzipped size 110kb. Submitted from: schlump@math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0910.3867 or http://arXiv.org/abs/0910.3867 or by email in unzipped form by transmitting an empty message with subject line uget 0910.3867 or in gzipped form by using subject line get 0910.3867 to: math@arXiv.org.