Taxidermy is the official term for stuffing. The word is composed of the Greek word for movement (taxis) and skin (derma) and refers to the technique of moving the skin to the location on the (artificial) body, where it belongs.
It is not easy to give the dead animal a live appearance. There are usually at least as many hours of shaping the animal as it takes (technically) to preserve it.
In natural history collections in museums, it is not common to use so much time, to give the animal a natural look, but animal skins, especially birds, are pulled over a stick with cotton wool.
It is special characteristics like, for example, the beak size, feather colour or paws length which are important parameters for research and here it is the durability that has top priority. It is only in exhibition copies where the natural posture comes into the picture.
Down here are some examples of my taxidermy work.
You find the latest examples here.