Invasions by alien plants provide a unique opportunity to examine competitive interactions among plants. While resource competition has long been regarded as a major mechanism responsible for successful invasions, given a well-known capacity for many invaders to become dominant and reduce plant diversity in the invaded communities, few studies have measured resource competition directly or have assessed its importance relative to that of other mechanisms, at different stages of an invasion process. Here, we review evidence comparing the competitive ability of invasive species vs. that of co-occurring native plants, along a range of environmental gradients, showing that many invasive species have a superior competitive ability over native species, although invasive congeners are not necessarily competitively superior over native congeners, nor are alien dominants are better competitors than native dominants. We discuss how the outcomes of competition depend on a number of factors, such as the heterogeneous distribution of resources, the stage of the invasion process, as well as phenotypic plasticity and evolutionary adaptation, which may result in increased or decreased competitive ability in both invasive and native species. Competitive advantages of invasive species over natives are often transient and only important at the early stages of an invasion process. It remains unclear how important resource competition is relative to other mechanisms (competition avoidance via phenological differences, niche differentiation in space associated with phylogenetic distance, recruitment and dispersal limitation, indirect competition, and allelopathy). Finally, we identify the conceptual and methodological issues characterizing competition studies in plant invasions, and we discuss future research needs, including examination of resource competition dynamics and the impact of global environmental change on competitive interactions between invasive and native species.