Adoption..

This may sound like a really stupid question but I am legitimately serious and curious. If someone just wants to give me their child, hand it over and have no rights to it what so ever, can they?! Is that illegal?! I really don't know how any of this works which is why I'm asking. Adoption is obviously really expensive for everyone involved so I'm just curious as to why if this is indeed actually possible, why more people don't do it? Maybe I'm just being crazy lol.