This is an announcement for the paper "Characterization of approximation schemes satisfying Shapiro's theorem" by J. M. Almira. Abstract: In this paper we characterize the approximation schemes that satisfy Shapiro's theorem and we use this result for several classical approximation processes. In particular, we study approximation of operators by finite rank operators and n-term approximation for several dictionaries and norms. Moreover, we compare our main theorem with a classical result by Yu. Brundyi and we show two examples of approximation schemes that do not satisfy Shapiro's theorem. Archive classification: math.CA math.FA The source file(s), almira_shapiro_theorem.tex: 47247 bytes, is(are) stored in gzipped form as 0910.2826.gz with size 14kb. The corresponding postcript file has gzipped size . Submitted from: jmalmira@ujaen.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0910.2826 or http://arXiv.org/abs/0910.2826 or by email in unzipped form by transmitting an empty message with subject line uget 0910.2826 or in gzipped form by using subject line get 0910.2826 to: math@arXiv.org.