This report describes a risk assessment of the alien species of Asian knotweeds in Europe. This species group comprises four species: Japanese knotweed (Reynoutria japonica), Bohemian knotweed (R. x bohemica), Giant knotweed (R. sachalinensis) and Himalayan knotweed (Koenigia polystachya). The species are native to Asia, but have been imported in many places in the world as ornamental species. They are all giant rhizomatous herbs, growing up to some meters high. The species are not only present in many European countries, but are often considered invasive as well. In Europe, in many cases these species have been introduced as vegetatively propagated ornamental plants. The present risk assessment is based on a detailed risk inventory and supports national and international decision making on the management of Asian knotweeds. The available information and data on the four species were analysed and the risks were classified by a team of experts using the Harmonia+ protocol.