Species invasions have become a major threat to global biodiversity. We currently lack a general theory of species invasions that allows us to make useful predictions about when and where invasions will occur, whether they will be successful, and whether they will alter ecosystem function in invaded habitats. One line of enquiry in developing such a theory is to focus on the characteristics of successful invaders. A second, complementary approach is to examine habitats of interest more closely and ask how the properties of the habitat that is being invaded affect the likelihood of invasion success. In this paper, I consider the importance of habitat shape (also termed "habitat topology" or "habitat geometry") as a variable affecting the dispersal and abundance of invasive populations. I use two well-established simulation modeling approaches, namely, a cellular automaton model and a reaction-diffusion model, to mimic species invasions in hypothetical habitats that cover a range of linear, branching, rectangular, and square shapes. The results suggest that invasions in more geometrically complex habitats will occur faster and may ultimately produce a higher abundance of the invasive species. Differences in invasion rates are not a simple consequence of differences in overall connectivity, as shown by a comparison of habitats with identical connectivities but different spatial arrangements of cells. Ultimately, if combined with other modeling approaches, these methods may be useful in generating recommendations for managers about the vulnerability of particular habitats and reserve networks to invasion.