Transposable elements (TEs) are powerful motors of genome evolution yet a comprehensive assessment of recent transposition activity at the species level is lacking for most organisms. Here, using genome sequencing data for 211 Arabidopsis thaliana accessions taken from across the globe, we identify thousands of recent transposition events involving half of the 326 TE families annotated in this plant species. We further show that the composition and activity of the 'mobilome' vary extensively between accessions in relation to climate and genetic factors. Moreover, TEs insert equally throughout the genome and are rapidly purged by natural selection from gene-rich regions because they frequently affect genes, in multiple ways. Remarkably, loci controlling adaptive responses to the environment are the most frequent transposition targets observed. These findings demonstrate the pervasive, species-wide impact that a rich mobilome can have and the importance of transposition as a recurrent generator of large-effect alleles.