Repeat elements are important components of eukaryotic genomes. One limitation in our understanding of repeat elements is that most analyses rely on reference genomes that are incomplete and often contain missing data in highly repetitive regions that are difficult to assemble. To overcome this problem we develop a new method, REPdenovo, which assembles repeat sequences directly from raw shotgun sequencing data. REPdenovo can construct various types of repeats that are highly repetitive and have low sequence divergence within copies. We show that REPdenovo is substantially better than existing methods both in terms of the number and the completeness of the repeat sequences that it recovers. The key advantage of REPdenovo is that it can reconstruct long repeats from sequence reads. We apply the method to human data and discover a number of potentially new repeats sequences that have been missed by previous repeat annotations. Many of these sequences are incorporated into various parasite genomes, possibly because the filtering process for host DNA involved in the sequencing of the parasite genomes failed to exclude the host derived repeat sequences. REPdenovo is a new powerful computational tool for annotating genomes and for addressing questions regarding the evolution of repeat families. The software tool, REPdenovo, is available for download at https://github.com/Reedwarbler/REPdenovo.