Transposases move discrete pieces of DNA between genomic locations and had a profound impact on evolution. They drove the emergence of important biological functions and are the most frequent proteins encoded in modern genomes. Yet, the molecular principles of their actions have remained largely unclear. Here we review recent structural studies of transposase-DNA complexes and related cellular machineries, which provided unmatched mechanistic insights. We highlight how transposases introduce major DNA twists and kinks at various stages of their reaction and discuss the functional impact of these astounding DNA acrobatics on several aspects of transposition. By comparison with distantly related DNA recombination systems, we propose that forcing DNA into unnatural shapes may be a general strategy to drive rearrangements forward.