This is an announcement for the paper "Disjointly homogeneous Banach lattices and applications" by Julio Flores, Francisco L. Hernandez and Pedro Tradacete. Abstract: This is a survey on disjointly homogeneous Banach lattices and their applicactions. Several structural properties of this class are analyzed. In addition we show how these spaces provide a natural framework for studying the compactness of powers of operators allowing for a unified treatment of well-known results. Archive classification: math.FA Mathematics Subject Classification: 47B38, 46E30 Remarks: 20 pages, to appear in Proceedings Positivity VII Conference Submitted from: ptradace@math.uc3m.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.01499 or http://arXiv.org/abs/1509.01499