This paper is intended to serve as an introduction to genome rearrangement and its use for inferring phylogenetic trees. We begin with a brief description of the major players of the field (chromosomes, genes, etc.) and the types of mutations that can affect them, focussing obviously on genome rearrangement. This leads to a simple mathematical representation of the data (the order of the genes on the chromosomes), and the operations that modify it (inversions, transpositions, and

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... . We then consider the problem of inferring phylogenetic (evolutionary) trees from genetic data. We briefly present the two major approaches to solve this problem. The first one, called distance matrix method, relies on the estimation of the evolutionary distance between each pair of species considered. In the context of gene order data, a useful measure of evolutionary distance is the minimum number of basic operations needed to transform the gene order of one species into that of another. This family of algorithmic problems has been extensively studied, and we review the major results in the field. The second approach to inferring phylogenetic trees consists of finding a minimal Steiner tree in the space of the data considered, whose leaves are the species of interest. This approach leads to much harder algorithmic problems. The main results obtained here are based on a simple evolutionary metric, the number of breakpoints. Throughout the paper, we report on various biological data analyses done using the different techniques discussed. We also point out some interesting open problems and current research directions.