Because of the way we think about adoption now, probably the first thing that springs to mind is a man without an heir who then adopts one. This did happen, but men who already had sons adopted more just as frequently, strengthening their chances of having descendants. Sometimes this was a way of legitimizing an illegitimate son, but that was less common. Usually they took legitimate sons of their own social class instead.
It was also unusual but not unheard of for a man to adopt a woman as his heir, as in the case of Curtilius Mancia, who hated his son-in-law and so adopted his granddaughter instead. Adoption allowed men to be highly selective about who became their heirs, but it also meant they weren’t just dependent on their offspring for security. When the Leges Iuliae came into effect, some childless men adopted others to take advantage of the benefits of having children and then emancipated them shortly thereafter.
There were benefits to the person being adopted as well, beyond just the chance at inheriting their stuff. Since adoption didn’t prevent him from inheriting from their birth parents, a young man adopted by another man of wealth could then inherit from two fathers instead of one. He might also gain the status of his new father. This led to several men of senatorial rank having themselves adopted by plebeians so they were eligible to become the Tribune of the Plebs. For Romans of the upper classes at least, adoption could be more of political strategy than anything else.
*Most forms of adoption weren’t open to women, and the wife of someone’s adoptive father was not considered to be one’s adoptive mother.
Corbier, Mireille. "Divorce and Adoption as Roman Familial Strategies." In Marriage, Divorce, and Children in Ancient Rome, edited by Beryl Rawson, 47-78. Oxford: Oxford University Press, 1991.
Adoptio - Lacus Curtius
Lindsay, Hugh. Adoption in the Roman World. Cambridge: Cambridge University Press, 2009.
Frier, Bruce. Review of Lindsay, Adoption in the Roman World - Bryn Mawr