I showed Rabbi Chaim Pinchus Scheinberg, shlita , an edition of Scrabble which had a flat board, without grooves for the tiles, and asked him if it is permitted to play this game on Shabbat. He acknowledged that some Poskim forbid it, and others permit it. He ruled that it is permitted, but that great care must be taken to ensure that the players do not forget that it's Shabbat and write down their scores.
Monopoly is a game that mimics business transactions.
Rabbi Scheinberg, shlita, says that technically it is permitted, but playing a business-oriented game on Shabbat is not conducive to a proper attitude about Shabbat. Chess is discussed in the Shulchan Aruch. The Rama mentions it as a pastime that is permitted on Shabbat. Some of the Poskim note, though, that in order to differentiate between Shabbat and weekday, some people had special Chess sets made of Silver just for Shabbat play. Chess clocks are another matter.