And keep in mind that when someone says "here" they are referring to the State that they live in. Bag limits, dates of hunting seasons, kinds of wildlife, all vary from state to state, or even region within a State. Here in Kentucky, we have Whitetail deer, but no Mule deer, or Blacktails. Elk are only in the mountains in the very eastern edge of Kentucky. No pheasants to speak of, but we have Mourning doves, ducks, geese, quail, and turkey that are the most popular game birds. Cottontail rabbits are plentiful, Jack rabbits live out West.
Of course this only scratches the surface, but its really interesting to hear about the differences of hunting here and the UK.
Btw, here in America, folks refer to pigeons as "flying" rats