Down syndrome (DS), also known as trisomy 21, is the most common genetic cause of intellectual disability (ID). Although ID can be mild, the average intelligence quotient is in the range of 40–50. All individuals with DS will also develop the neuropathology of Alzheimer’s disease (AD) by the age of 30–40 years, and approximately half will display an AD-like dementia by the age of 60 years. DS is caused by an extra copy of the long arm of human chromosome 21 (Hsa21) and the consequent elevated levels of expression, due to dosage, of trisomic genes. Despite a worldwide incidence of one in 700–1,000 live births, there are currently no pharmacological treatments available for ID or AD in DS. However, over the last several years, very promising results have been obtained with a mouse model of DS, the Ts65Dn. A diverse array of drugs has been shown to rescue, or partially rescue, DS-relevant deficits in learning and memory and abnormalities in cellular and electrophysiological features seen in the Ts65Dn. These results suggest that some level of amelioration or prevention of cognitive deficits in people with DS may be possible. Here, we review information from the preclinical evaluations in the Ts65Dn, how drugs were selected, how efficacy was judged, and how outcomes differ, or not, among studies. We also summarize the current state of human clinical trials for ID and AD in DS. Lastly, we describe the genetic limitations of the Ts65Dn as a model of DS, and in the preclinical testing of pharmacotherapeutics, and suggest additional targets to be considered for potential pharmacotherapies.