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Realizing the democratic promise of nanopore sequencing requires the development of new bioinformatics approaches to deal with its specific error characteristics. Here we present GraphMap, a mapping algorithm designed to analyse nanopore sequencing reads, which progressively refines candidate alignments to robustly handle potentially high-error rates and a fast graph traversal to align long reads with speed and high precision (>95%). Evaluation on MinION sequencing data sets against short- and long-read mappers indicates that GraphMap increases mapping sensitivity by 10-80% and maps >95% of bases. GraphMap alignments enabled single-nucleotide variant calling on the human genome with increased sensitivity (15%) over the next best mapper, precise detection of structural variants from length 100 bp to 4 kbp, and species and strain-specific identification of pathogens using MinION reads. GraphMap is available open source under the MIT license at https://github.com/isovic/graphmap.