This article reviews recent developments in Bayesian algorithms that explicitly include geographical information in the inference of population structure. Current models substantially differ in their prior distributions and background assumptions, falling into two broad categories: models with or without admixture. To aid users of this new generation of spatially explicit programs, we clarify the assumptions underlying the models, and we test these models in situations where their assumptions are not met. We show that models without admixture are not robust to the inclusion of admixed individuals in the sample, thus providing an incorrect assessment of population genetic structure in many cases. In contrast, admixture models are robust to an absence of admixture in the sample. We also give statistical and conceptual reasons why data should be explored using spatially explicit models that include admixture.