The early Romans had a good policy on the exposition of children. According to Dionysius of Halicarnassus, Romulus imposed on all citizens the obligation to raise all male children and the eldest of girls.  If the children were deformed and monstrous, he allowed them to be exposed, after showing them to five of one’s closest neighbors.
Romulus did not permit the killing of any child under three years of age  ; that way he reconciled the law that gave fathers the right of life and death over their children and the one that forebade exposing them.
We further find in Dionysius of Halicarnassus that the law which ordered citizens to marry and raise all their children was in force in the Roman year 277 ; we see that custom had restrained the law of Romulus, which allowed the exposition of younger daughters. 
We are aware of what the law of the Twelve Tables, issued in the year of Rome 301, decreed about the exposition of children, only from a passage of Cicero who, speaking of the tribunate of the people, says that immediately after its birth, like the monstrous child of the law of the Twelve Tables, it was smothered  : children who were not monstrous were therefore preserved, and the law of the Twelve Tables changed nothing about the previous institutions.
“The Germans,” says Tacitus, “do not expose their children, and for them good practices are more influential than good laws are elsewhere.”  Therefore the Romans did have laws against this custom, and they were no longer followed. We find no Roman law  that allows the exposition of children ; it was doubtless an abuse introduced in the latter times, when luxury replaced plenty, when shared wealth was called poverty, when the father considered as lost what he gave to his family, and distinguished that family from his property.