Infanticide and Exposure
From the Sumerians down to the time of Augustine, most people agreed that if you had a newborn baby that you didn't want, you could kill it or abandon it somewhere and not take care of it. Killing a baby is called infanticide, and leaving it somewhere is called exposure.
Sometimes, the father decided whether to keep a baby or not. Of course most fathers wanted to keep their babies, and most of the time they did, but sometimes they felt it was not the right thing to do. Sometimes a baby would be exposed if it seemed to be sick or for some reason was not likely to live anyway. If there was something seriously wrong with the baby's heart or stomach or something, then ancient people could not do surgery to fix that. The baby would probably die anyway in a few days.
Other babies were exposed because they were physically
challenged: they couldn't see or hear, or they were born with only
one arm or one leg. Maybe their parents felt they couldn't take care
of them. (But other physically challenged babies were raised).
Sometimes if there were twins, one of the two babies would be exposed, because people thought twins were unlucky. Sometimes people abandoned girl babies, because they wanted a boy.
In very poor families (but there were a lot of very poor families in the ancient world), sometimes babies were exposed because the family didn't have enough food to feed everyone. They might decide it was better to give the food to the older kids than to split it among everyone if there wasn't enough.
Other babies were abandoned by other people, not their parents. For slaves, their owner decided whether to keep them or abandon them. Even some free people were so much under the power of the government or a rich patron that someone else would decide for them. In ancient Sparta, for example, the government decided whether you could keep your baby.
Most of the babies who were exposed probably died, but some of them were picked up by people who couldn't have babies themselves and wanted to adopt a baby. Others were picked up by slave dealers and raised to be sold as slaves. Mothers and fathers who had to expose their babies always wanted to think that their baby had been saved (this is the plot of many Greek and Roman plays).
BILL OF RIGHTS
WHAT IS BC OR AD?
After most people in Europe converted to Christianity, around 400 AD, infanticide and exposure became a little less common, because the Church said that babies had souls too and so it was murder to kill them or let them die. Still this did not stop many people from leaving babies on the steps of churches for somebody else to take care of, and these abandoned babies often also died.