In a north European context, identifying invasive species is far from trivial because the vegetation has been influenced by human activities for thousands of years. New species have immigrated continuously since the end of the last glaciation, both spontaneously and through direct or indirect help by man. Still, newly immigrated species may be problematic in a nature conservation context and may harm biodiversity, and cause changes in ecosystem services. In this study, all 721 presently established vascular plant and bryophyte taxa known to have been introduced to, or to have immigrated to, Sweden since the year 1700 are assessed for their invasive potential. The assessment is based on six components considered relevant for their invasiveness: 1) ability to compete in natural vegetation, 2) ability to form dense populations, 3) realized dispersal ability, 4) gene flow to native relatives, 5) time since immigration and 6) distance to their native range. Although the relative importance of these components is context dependent, a compound general index of invasive concern is proposed and calculated for all taxa. The index of invasive concern is validated by comparing it to assessments by a Bayesian Belief Network in which the influences of the same six components are assigned by expert opinions. In addition, rough estimates of the present-day abundance of each taxon in Sweden are presented. The origin, biology and possible impact of the 150 taxa with the highest index values are discussed. Based on the index of invasive concern, Campylopus introflexus, Epilobium adenocaulon, E. ciliatum, Rosa rugosa, Lamiastrum galeobdolon subsp. argentatum, Orthodontium lineare, Solidago canadensis, Calystegia sepium subsp. spectabilis, Rubus armeniacus and Prunus serotina are identified as the ten most problematic alien vascular plant and bryophyte taxa in Sweden.