Cat's which have been abandoned by their owners or born and brought up on the streets carve themselves a life as strays. Within a short period of time, they revert from domesticated cats to being wild (feral) cats, living and forming colonies with other cats. They also breed prolifically.
It is perfectly possible to adopt a stray or feral cat directly off the streets and you might find that the choice was not you adopting it, rather the other way round!
You have to be careful of a few things though because they are exposed to a host of infections and diseases on the streets and so it's important to get it some medical attention before too much contact.
Obviously, spending time with your new cat is important because the bond needs to develop to help it adjust to a new settled way of life.