Recent methodological advances have enabled Maximum Likelihood estimation of the ancestral ranges of organisms, given a phylogenetic topology, information on divergence times, and terminal geographic ranges. Such approaches, as implemented in the R package BioGeoBEARS, allow different models of biogeographic evolution to be fitted to the data and compared statistically. This permits ancestral ranges and the relative importance of processes such as dispersal and vicariance to be estimated. Here, BioGeoBEARS is applied to five data sets for dinosaurs (macronarian and diplodocoid sauropods, tyrannosaurs, Abelisaurs and allosaurs), with and without constraints based on the changing paleogeography of Pangaea through the Mesozoic. Results indicate that: 1. Previous work based on the application of the Dispersal-Extinction-Cladogenesis (DEC) method to fossil data sets are problematic because of the lack of testing of competing models and the assumption that phylogenies are ultrametric; 2. Analyses that are not constrained by paleogeographic data are dominated by founder-event speciation as the explanation of ancestral distributions, which seems improbable for continental taxa; and 3. Paleogeographically constrained analyses reduce the role for founder-event speciation and increase the roles of regional extinction and/or vicariance. Paleogeographic constraints therefore form an important component in this type of biogeographic analysis, but are subject to problems caused by uncertainties in the sequence and timing of area coalescence and fragmentation events. A simple protocol for capturing these uncertainties and using them in sensitivity analyses in BioGeoBEARS is proposed. The results from the dinosaur data indicate that, although the exact details of estimated biogeographic histories are sensitive to alternative paleogeographic constraints, there is general consensus across most analyses. There remain many significant challenges to the use of Maximum Likelihood approaches to paleobiogeography, not least of which includes accurate dating of divergence times in clades that have no living descendants, sampling failures in the fossil record, and the incompleteness or inaccuracy of phylogenetic trees.