Hunting with air guns has a long history thats well documented since the mid-18th century. In 1747 Louis VIII, Landgrave of Hesse, killed a 480-pound stag with an air gun, and another member of the court took a stag at 154 paces! While you're unlikely to hunt red deer with an air gun, the excitement of the chase is as close as a nearby brush pile or woodlot.
Where its legal, ruffled grouse, cottontail rabbits and fox squirrels can test your skills with an air gun. You could try various ground squirrels for a challenge, and marmots and jackrabbits aren't too tough for some .22 air guns. It's important to remember that a 40-yard shot is a stretch even for the most powerful air guns. Hunting with air pistols is essentially a 15-yard challenge and its really unethical to push that limit without knowing that you and your equipment are clearly capable of more.
The short effective range of air guns, and their quiet manner, makes them acceptable in some situations where firearms would not be, such as near livestock and farmsteads and in fencerows bordering subdivisions, among others.
(Material courtesy of National Shooting Sports Foundation)