Down syndrome (DS) is caused by trisomy of all or part of human chromosome 21 (HSA21) and is the most common genetic cause of significant intellectual disability. In addition to intellectual disability, many other health problems, such as congenital heart disease, Alzheimer’s disease, leukemia, hypotonia, motor disorders, and various physical anomalies occur at an elevated frequency in people with DS. On the other hand, people with DS seem to be at a decreased risk of certain cancers and perhaps of atherosclerosis. There is wide variability in the phenotypes associated with DS. Although ultimately the phenotypes of DS must be due to trisomy of HSA21, the genetic mechanisms by which the phenotypes arise are not understood. The recent recognition that there are many genetically active elements that do not encode proteins makes the situation more complex. Additional complexity may exist due to possible epigenetic changes that may act differently in DS. Numerous mouse models with features reminiscent of those seen in individuals with DS have been produced and studied in some depth, and these have added considerable insight into possible genetic mechanisms behind some of the phenotypes. These mouse models allow experimental approaches, including attempts at therapy, that are not possible in humans. Progress in understanding the genetic mechanisms by which trisomy of HSA21 leads to DS is the subject of this review.