It is narrated from the prophet that,
"When you release your arrow or your dog, mentioned the name of Allah, and when your arrow kills (the game), then eat." ['Adiyy bin Hatim] said, "What if it gets away from me for a night, O Messenger of Allah?" He said: "If you find your arrow and you do not find the mark of anything else, then eat it. But if it falls into the water, do not eat it."
The prohibition inre the water is due to the fact that if the animal is found in the water, it is not known whether it was the arrow that killed it or if it died due to the water (e.g. drowning). Basically, as long as you speak bismillah on your weapon (arrows and dogs are mentioned, being the primary hunting tools at the time; the same reasoning can easily apply to guns and other modern weapons), and it is known that that weapon is what killed the animal, you may eat of it.
The basic rulings of dhabihah would still need to be adhered to, particularly (but not limited to) that the animal does not die by bludgeoning or strangling, and the blood would still need to be removed. Ergo, your weapon should be one which kills by piercing or cutting the animal (which would cause blood to flow); even if you say bismillah over a bludgeoning weapon before striking, any animal that dies therefrom would not be halal.
An extra consideration when using a hunting animal is that even if you say bismillah before releasing him, you may only eat of the prey if the animal hasn't eaten of it. This is based on a different hadith which states "if the dog eats of (that game) then do not eat it because the dog has hunted it for itself" rather than hunting it for you.
If the hunted animal is wounded rather than killed, such that you are able to find and slaughter it appropriately, then no special considerations need to be made beyond the regular requirements of dhabihah.