This is an announcement for the paper "Almost overcomplete and almost overtotal sequences in Banach spaces" by Vladimir P. Fonf and Clemente Zanco. Abstract: The new concepts are introduced of almost overcomplete sequence in a Banach space and almost overtotal sequence in a dual space. We prove that any of such sequences is relatively norm-compact and we obtain several applications of this fact. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B50, 46B45 Submitted from: clemente.zanco@unimi.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1402.6247 or http://arXiv.org/abs/1402.6247