Their ability to move within genomes gives transposable elements an intrinsic propensity to affect genome evolution. Non-long terminal repeat (LTR) retrotransposons — including LINE-1, Alu and SVA elements — have proliferated over the past 80 million years of primate evolution and now account for approximately one-third of the human genome. In this Review, we focus on this major class of elements and discuss the many ways that they affect the human genome: from generating insertion mutations and genomic instability to altering gene expression and contributing to genetic innovation. Increasingly detailed analyses of human and other primate genomes are revealing the scale and complexity of the past and current contributions of non-LTR retrotransposons to genomic change in the human lineage.