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Listed below are websites that have a wide selection of board games that you can play. Most of the board games that are available on these websites are somewhat more complicated strategy games, but they include rules on how to play them.
These websites allow for either asynchronous play (i.e., turn-based play, with the ability to wait days between moves) or live play, or both. These websites require creating an account to play, and unless otherwise noted, are free.