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Abstract Aim The tomato family Solanaceae is distributed on all major continents except Antarctica and has its centre of diversity in South America. Its worldwide distribution suggests multiple long‐distance dispersals within and between the New and Old Worlds. Here, we apply maximum likelihood ( ML ) methods and newly developed biogeographical stochastic mapping ( BSM ) to infer the ancestral range of the family and to estimate the frequency of dispersal and vicariance events resulting in its present‐day distribution. Location Worldwide. Methods Building on a recently inferred megaphylogeny of Solanaceae, we conducted ML model fitting of a range of biogeographical models with the program ‘BioGeo BEARS ’. We used the parameters from the best fitting model to estimate ancestral range probabilities and conduct stochastic mapping, from which we estimated the number and type of biogeographical events. Results Our best model supported South America as the ancestral area for the Solanaceae and its major clades. The BSM analyses showed that dispersal events, particularly range expansions, are the principal mode by which members of the family have spread beyond South America. Main conclusions For Solanaceae, South America is not only the family's current centre of diversity but also its ancestral range, and dispersal was the principal driver of range evolution. The most common dispersal patterns involved range expansions from South America into North and Central America, while dispersal in the reverse direction was less common. This directionality may be due to the early build‐up of species richness in South America, resulting in large pool of potential migrants. These results demonstrate the utility of BSM not only for estimating ancestral ranges but also in inferring the frequency, direction and timing of biogeographical events in a statistically rigorous framework.