Genomics has revolutionised the study of invasive species, allowing evolutionary biologists to dissect mechanisms of invasion in unprecedented detail. Botanical research has played an important role in these advances, driving much of what we currently know about key determinants of invasion success (e.g. hybridisation, whole-genome duplication). Despite this, a comprehensive review of plant invasion genomics has been lacking. Here, we aim to address this gap, highlighting recent discoveries that have helped progress the field. For example, by leveraging genomics in natural and experimental populations, botanical research has confirmed the importance of large-effect standing variation during adaptation in invasive species. Further, genomic investigations of plants are increasingly revealing that large structural variants, as well as genetic changes induced by whole-genome duplication such as genomic redundancy or the breakdown of dosage-sensitive reproductive barriers, can play an important role during adaptive evolution of invaders. However, numerous questions remain, including when chromosomal inversions might help or hinder invasions, whether adaptive gene reuse is common during invasions, and whether epigenetically induced mutations can underpin the adaptive evolution of plasticity in invasive populations. We conclude by highlighting these and other outstanding questions that genomic studies of invasive plants are poised to help answer.