Transposable elements (TEs) are discrete genetic units that have the ability to change their location within chromosomal DNA, and constitute a major and rapidly evolving component of eukaryotic genomes. They can be subdivided into 2 distinct types: retrotransposons, which use an RNA intermediate for transposition, and DNA transposons, which move only as DNA. Rapid advances in genome sequencing significantly improved our understanding of TE roles in genome shaping and restructuring, and studies of transcriptomes and epigenomes shed light on the previously unknown molecular mechanisms underlying genetic and epigenetic TE controls. Knowledge of these control systems may be important for better understanding of reticulate evolution and speciation in the context of bringing different genomes together by hybridization and perturbing the established regulatory balance by ploidy changes. See also sister article focusing on plants by Bento et al. in this themed issue.