Transposable elements are known by many names, including 'transposons', 'interspersed repeats', 'selfish genetic elements', 'jumping genes', and 'parasitic DNA', but here we will refer to them simply as transposable elements. Many biologists will have heard of transposable elements and their ability to transpose (change position) within the genome. But fewer may be aware of their varied influences on host biology, including contributions to the evolution of diverse host traits such as internal gestation, memory, colouration, and adaptive immunity. Transposable elements are a near ubiquitous feature of eukaryotic genomes, and they often comprise a substantial proportion of total genomic content. Consequently, transposable element genes are considered among the most abundant coding sequences in nature. Recent advances in genome sequencing have ushered in a golden age for transposable-element research, providing opportunities to greatly improve our understanding of the effects of transposable elements on host evolution and disease. However, our ability to detect and analyse transposable elements still faces significant challenges, impairing efforts to decipher their evolution, characterise their diversity, and elucidate their myriad host influences. Below, we summarise key aspects of transposable element biology in eukaryotes and discuss major outstanding research questions.