This describes the rules of the game Barbu as played at Yale (and spread thence to Maine by Joel Bradford).
See Nick Wedd's page for a nice overview and pointers to other Barbu pages around the web. If you're already familiar with those rules, jump to the the page of differences from Wedd's Barbu.
Barbu is a game of seven sub-games. Each player choses one of the games to play once. If there are four players (as for the canonical game), this leads to twenty eight hands. The games are Barbu, Queens, Hearts, Trumps, Nullo, Last two and Fan tan.
All the games except Fan tan are played with bridge tricks. Fan tan is played as for standard Fan tan, except that the dealer (who also chooses the game) gets to choose the start card (i.e. it is not always 7). There is no trump suit for the other hands except in Trumps, where the trump is also chosen by the dealer (who can also choose to play this with no trump).
For Barbu, hearts may not be led unless the player on the lead has nothing but hearts. For Hearts, hearts may not be led unless the player has nothing but hearts, or hearts have been "broken", i.e. a heart has been thrown off or lead. Additional rules apply to players who aren't doubled, described below.
After dealer has chosen the game and options (i.e. the start card for Fan tan and the trump suit for Trumps), each player in turn (in play order) can choose whether to double the dealer. After the doubles are made, the dealer can choose whether to redouble each player individually. If a there is a double between two players, then the difference between their scores is added to the score of the player who got a higher (or less negative) score and the same amount is subtracted from the loser's score. If they are redoubled, then twice that difference is added and subtracted.
For example, assume dealer chooses Queens, is doubled by all players, and redoubles the player on his right. Dealer takes one queen, the player on dealer's right takes two, and the player on dealer's left takes one. The starting points are (starting with dealer's left), -6, 0, -12, -6. Since the dealer and dealer's left have the same score, they don't exchange points: dealer's left scores -6. The next player did better than dealer, she gets the difference added to her score and subtracted from his, so she scores +6. Dealer's right does worse than dealer by 6, and this is redoubled, so that player looses 12 more and dealer gains that; that player scores -18. Dealer started with -6, lost 6 to the across player, and gained 12 from dealer's right, so her net score gain is 0.
To improve fairness, some special rules of play apply to players who have not doubled the dealer, and to the dealer if undoubled.
Fan tan is more interesting when it can wrap around, i.e. have an ace be playable on a king and vice versa.
You need to keep track of which games each player has chosen, and allow space for the arithmetic for calculating the score adjustments for doubles. This spreadsheet has a printable score sheet on one tab, a tab that will calculate the sums for you, and a tab that will calculate the scores per hand. With that fancier tab, you don't need to remember what the score for each thing is, nor do you need to do the doubles arithmetic. It also includes a checksum to make sure you got all the points which should be scored per hand.