In comparative genomics, a transposition is an operation that ex- changes two consecutive sequences of genes in a genome. The transposition distance, that is, the minimum number of transpositions needed to transform a genome into another, can be considered as a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permuta- tions, called the S ORTING BY T RANSPOSITIONS problem (SBT), has been in- troduced by Bafna and Pevzner [3] in 1995. It has naturally been the focus of a number of studies, but the computational complexity of this problem has re- mained undetermined for 15 years. In this paper, we answer this long-standing open question by proving that the S ORTING BY T RANSPOSITIONS problem is NP-hard. As a corollary of our re- sult, we also prove that the following problem from [9] is NP-hard: given a per- mutation π, is it possible to sort π using db (π)/3 permutations, where db (π) is the number of breakpoints of π?