Invasion rates have increased in the past 100 y irrespective of international conventions. What characterizes a successful invasion event? And how does genetic diversity translate into invasion success? Employing a whole-genome perspective using one of the most successful marine invasive species world-wide as a model, we resolve temporal invasion dynamics during independent invasion events in Eurasia. We reveal complex regionally independent invasion histories including cases of recurrent translocations, time-limited translocations, and stepping-stone range expansions with severe bottlenecks within the same species. Irrespective of these different invasion dynamics, which lead to contrasting patterns of genetic diversity, all nonindigenous populations are similarly successful. This illustrates that genetic diversity, per se, is not necessarily the driving force behind invasion success. Other factors such as propagule pressure and repeated introductions are an important contribution to facilitate successful invasions. This calls into question the dominant paradigm of the genetic paradox of invasions, i.e., the successful establishment of nonindigenous populations with low levels of genetic diversity.