The appearance of cognitive deficits and altered brain morphology in newborns with Down syndrome (DS) suggests that these features are driven by disruptions at the earliest stages of brain development. Despite its high prevalence and extensively characterized cognitive phenotypes, relatively little is known about the cellular and molecular mechanisms that drive the changes seen in DS. Recent technical advances, such as single-cell omics and the development of induced pluripotent stem cell (iPSC) models of DS, now enable in-depth analyses of the biochemical and molecular drivers of altered brain development in DS. Here, we review the current state of knowledge on brain development in DS, focusing primarily on data from human post-mortem brain tissue. We explore the biological mechanisms that have been proposed to lead to intellectual disability in DS, assess the extent to which data from studies using iPSC models supports these hypotheses, and identify current gaps in the field.