To tackle this problem, a new alignment tool BWBBLE is introduced in this article. We (i) introduce a new compressed representation of a collection of genomes, which explicitly tackles the genomic variation observed at every position, and (ii) design a new alignment algorithm based on the Burrows-Wheeler transform that maps short reads from a newly sequenced genome to an arbitrary collection of two or more (up to millions of) genomes with high accuracy and no inherent bias to one specific genome.