Closely related to the wild wood pigeon (colombaccio), the grey-breasted columba livia or common pigeon is the species most likely to be raised for food. (Though many game fanciers claim the colombaccio is tastier)
Whichever, it's a real treat in Italy, and surprisingly versatile. Roast it, stew it in salmì, stuff ravioli with it, put it in risottos or over pasta. Bake it in a tart with chanterelle mushrooms (tortelli al piccione ai finferli). Pigeon breast by itself is a common main course in restaurants, and you may even encounter pigeon carpaccio. Common flavourings in pigeon recipes are olives, garlic and rosemary.
In Umbria, Tuscany and Lazio, the area most known for pigeon, it will most likely be roasted in the oven or on a spit (alla leccarda). Usually they leave the heart, liver and lungs in, or make them into a stuffing with bread crumbs and onions.