Repetitive sequences make up a significant fraction of almost any genome, and an important and still open question in bioinformatics is how to represent all repeats in DNA sequences. We propose a new approach to repeat classification that represents all repeats in a genome as a mosaic of sub-repeats. Our key algorithmic idea also leads to new approaches to multiple alignment and fragment assembly. In particular, we show that our FragmentGluer assembler improves on Phrap and ARACHNE in assembly of BACs and bacterial genomes.