In early Rome, according to Philippe Ariès:
The birth of a Roman was not merely a biological fact. Infants came into the world, or at any rate were received into society, only as the head of the family willed. Contraception, abortion, the exposure of freeborn infants, and infanticide of slaves’ children were common and perfectly legal practices. They would not meet with disapproval or be declared illegal until a new morality had taken hold, a morality which for the sake of brevity I shall describe simply as Stoic. A citizen of Rome did not “have” a child; he “took” a child, “raised” him up (tollere). Immediately after the birth it was the father’s prerogative to raise the child from the earth where the midwife had placed it, thus indicating that he recognized the infant as his own and declined to expose it. […]
A child whose father did not raise it up was exposed outside the house or in some public place. Anyone who wished might claim it. An absent father might order his pregnant wife to expose her baby as soon as it was born. The Greeks and the Romans thought it peculiar that Egyptians, Germans, and Jews exposed none of their children but raised them all. In Greece it was more common to expose female infants than males. In 1 B.C. a Greek wrote his wife: “If (touch wood!) you have a child, let it live if it is a boy.If it is a girl, expose it.” It is not at all clear, however, that the Romans shared this prejudice. They exposed or drowned malformed infants. This, said Seneca, was not wrath but reason: “What is good must be set apart from what is good for nothing.” The Romans also exposed the children of their daughters who had “gone astray.” (Ariès et al. 9–10).
Ariès, Philippe, et al. A History of Private Life, Volume I: From Pagan Rome to Byzantium. Eds. Philippe Aries and Georges Duby. 12th ed. Cambridge, MA: Harvard University Press, 1992. Print.