Down syndrome (DS) is the most common genetic disease at birth; on average, it affects 1 in 700 newborns. The syndrome features cognitive impairment, susceptibility to certain diseases, and (in some cases) congenital malformations. Improvements in medical care for people with DS have led to an increase in life expectancy. Furthermore, the systematic provision of specific support during childhood improves cognitive function and autonomy in adulthood. Consequently, patients and their families are now seeking the same rights as healthy people. Access to procreation is an emerging debate. The presumption of infertility in DS is based on a few old studies. Down syndrome appears to cause spermatogenesis defects in men and premature menopause in women. When assisted reproductive technology makes it possible to solve these problems, the question of fertility in DS must be addressed. Without entering into highly controversial ethical considerations related to parenthood for people with DS, we reviewed the literature on fertility in DS and tried to specify the associated genetic risk.