There is growing realisation that integrating genetics and ecology is critical in the context of biological invasions, since the two are explicitly linked. So far, the focus of ecological genetics of invasive alien species (IAS) has been on determining the sources and routes of invasions, and the genetic make-up of founding populations, which is critical for defining and testing ecological and evolutionary hypotheses. However an ecological genetics approach can be extended to investigate questions about invasion success and impacts on native, recipient species. Here, we discuss recent progress in the field, provide overviews of recent methodological advances, and highlight areas that we believe are of particular interest for future research. First, we discuss the main insights from studies that have inferred source populations and invasion routes using molecular genetic data, with particular focus on the role of genetic diversity, adaptation and admixture in invasion success. Second, we consider how genetic tools can lead to a better understanding of patterns of dispersal, which is critical to predicting the spread of invasive species, and how studying invasions can shed light on the evolution of dispersal. Finally, we explore the potential for combining molecular genetic data and ecological network modelling to investigate community interactions such as those between predator and prey, and host and parasite. We conclude that invasions are excellent model systems for understanding the role of natural selection in shaping phenotypes and that an ecological genetics approach offers great potential for addressing fundamental questions in invasion biology.