Assuming that faster-than-light travel is physically impossible, humanity's options for colonizing other star systems are limited. A multi-generation starship or one in which the settlers are traveling in some form of suspended animation both entail a one-way trip for the colonists. The settlers are on their own, starting a new civilization.
If that civilization is going to be healthy, it's going to need genetic diversity in order to avoid the problems that come with inbreeding. In 2002, an anthropologist named John Moore determined that such a colony would need at least 150 people in order to maintain a healthy genetic diversity.
But Cameron Smith, an anthropologist at Portland State University, says that Moore's number is way too low. He and a colleague named William Gardner-O'Kearney created mathematical models to project the viability of different populations. Smith concluded that in order to be genetically healthy in the long run, a human space colony would need at least 10,000 members.
Ideally, though, Smith would suggest a colony of 40,000 humans to be on the safe side. This number would also protect the colony's genetic diversity if it lost significant numbers during the interstellar journey or early colonization period. Sarah Fecht explains in Popular Mechanics:
A starting population of 40,000 people maintains 100 percent of its variation, while the 10,000-person scenario stays relatively stable too. So, Smith concludes that a number between 10,000 and 40,000 is a pretty safe bet when it comes to preserving genetic variation. […]
"With 10,000," Smith says, "you can set off with good amount of human genetic diversity, survive even a bad disease sweep, and arrive in numbers, perhaps, and diversity sufficient to make a good go at Humanity 2.0."
-via Glenn Reynolds