The traditional approach for unowned cats—catching and euthanizing—is endless and cruel, and it does not keep an area free of cats.
Cats choose to reside in a location for two reasons: there is a food source (intended or not) and shelter. Because of a phenomenon called the vacuum effect, when cats are removed from a location, new cats move in to take advantage of the food resources and shelter, then breed to capacity.
Many unowned cats are not socialized to people and cannot be touched, except sometimes by a regular caregiver. The ideal window for socializing unowned kittens is 12 weeks of age or younger—beyond 12 weeks, unowned cats may never socialize completely or at all. Outdoor cats that are friendly and socialized to people are called stray cats, and sometimes they can be re-homed.
Because most unowned cats are not socialized to people, they are unadoptable as pets. In most shelters and pounds in the US, unadoptable animals are euthanized. If a cat exhibits signs of being feral- and are surrendered to a shelter instead of being trapped, neutered or spayed, then returned to the area where they were found- they will most likely be euthanized.
Many shelters now realize that Trap-Neuter-Return is the humane approach for free-roaming, unowned cats.