Wiktionary
n. The hunting of deer by stalking, or stealing upon them unawares. One who deer stalks is known as a deer stalker.
Wikipedia
Deer stalking is a British term for the stealthy pursuit of deer on foot with intention of killing the deer for meat, for sport, or to control the numbers. As part of a land management programme, just as with bird hunting and shooting, the aim with deer stalking is to reduce crop damage and to obtain food. Also, as with hunting, deer stalking has long been considered a sport. Stalked deer are commonly shot with a high powered bow (in countries other than the UK; in the UK it is illegal to hunt any animal or bird with a bow) or centre-fire rifle; prior to the invention of the modern rifle deer were stalked with the aid of a sighthound the Scottish Deerhound. Stalking sticks are often used to steady the aim of the rifle and to steady the binoculars when scanning the ground.
The term "deer hunting" is used in North America for the shooting of deer, whilst in Britain and Ireland the term generally refers to the pursuit of deer with scent hounds, with unarmed followers typically on horseback.