The BBC reports that the boxes - which have baby blankets and a letter of advice for the mother - have faced criticisms from the UN who claim they violate the rights of the infant.
The boxes were common in medieval Europe, and the modern equivalent - a heated hatch monitored by nurses - has the support of many agencies who claim it is better for the baby than the alternatives, including abandonment on the street.
The BBC describes one of the drop off points as being a metal hatch at the end of a path marked "Babywiege" - meaning Baby Cradle - in Germany.
Critics claim that some mums might be forced to abandon their babies against their will, and that sometimes it is not even mums putting their babies in the boxes, but pimps, fathers, other relatives or step-fathers.
Psychologist, Kevin Browne of Nottingham University told the BBC that a study undertaken in Hungary backed up this claim, and questioned whether the baby boxes were even 'upholding women's rights', asking: "Has the mother of that child consented to the baby being placed in the baby box?"
Professor Browne also added that the anonymity of the baby hatch was 'so removed from the availability of counselling' that it created 'a damage and a danger to the mother and child.'
Advocates of the scheme, however, say they are offering desperate mums a safe way to pass their unwanted babies on to a new home.
What do you think?
A custom that should have stayed in medieval times, or a safe option for desperate women?
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