This is an announcement for the paper "Lebesgue-Type inequalities for quasi-greedy bases" by Eugenio Hernandez. Abstract: We show that for quasi-greedy bases in real Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N-term error of approximation times a constant which depends on the democracy functions and the quasi-greedy constant of the basis. Archive classification: math.FA Mathematics Subject Classification: 41A65, 41A46, 41A17 Remarks: 8 pages Submitted from: eugenio.hernandez@uam.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1111.0460 or http://arXiv.org/abs/1111.0460