In Islam, it is not permissible to play backgammon because it involves the use of dice which are haram and emphatically forbidden. For more, see the detailed answer.

There are games which are specifically forbidden in the texts such as playing with dice. There are games which are neither enjoined nor forbidden in the texts and they are of two types: 1- The games which include something that is haram such as games that involve statues or images of living beings, or which are accompanied by music. 2- The games which do not involve anything haram, and which usually do not lead to it, like most of the games we see of football (soccer), volleyball and table-tennis etc.