A quantum state which can be written as a convex sum of product states is a separable state; such a convex sum allows for a local description. There are many different local descriptions for a given separable state, and some new results about the minimal number of product states in the convex sum wil be given in this paper. Any quantum state is either entangled or separable. However it is very difficult to tell whether a given state is separable or entangled. This difficulty has not been solved yet although a lot of separability criteria have already be proposed. The various separability criteria will also be discussed.