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Palmer amaranth (Amaranthus palmeri) is an annual plant native to the desert Southwest of the United States and Mexico and has become invasive and caused large economic losses across much of the United States. In order to examine the temporal and spatial dynamics of past invasion, and to predict future invasion, we developed a broad array of species distribution models (SDMs). In particular, we constructed sequential SDMs throughout the invasion history and asked how well those predicted future invasion (1970 to present). We showed that invasion occurred from a restricted set of environments in the native range to a diverse set in the invaded range. Spatial autocorrelation analyses indicated that rapid range expansion was facilitated by stochastic, long-distance dispersal events. Regardless of SDM approach, all SDMs built using datasets from early in the invasion (19702010) performed poorly and failed to predict most of the current invaded range. Together, these results suggest that climate is unlikely to have influenced early stages of range expansion. SDMs that incorporated data from the most recent sampling (2011-2017) performed considerably better, predicted high suitability in regions that have recently become invaded, and identified mean annual temperature as a key factor limiting northward range expansion. Under future climates, models predicted both further northward range expansion and significantly increased suitability across large portions of the U.S. Overall, our results indicate significant challenges for SDMs of invasive species far from climate equilibrium. However, our models based on recent data make more robust predictions for northward range expansion of A. palmeri with climate change.