None.
The Bengals have played in two Super Bowls and lost both. In Super Bowl XVI they lost to the San Francisco 49ers 26-21 and in Super Bowl XXIII they again lost to the 49ers 20-16.
The Cincinnati Bengals are 0-and-2 in Super Bowl appearances, having lost twice to the San Francisco 49ers.
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The Cincinnati Bengals have never won the Super Bowl. They have played in two of them, XVI (1981) and XXIII (1988), both of which they lost to the San Francisco 49ers (26-21, 20-16 respectively).
The Cincinnati Bengals have been to the playoffs 8 times: 1970, 1973, 1975, 1981, 1982, 1988, 1990, 2005
They appeared in two Super Bowls XVI (1981) and XXIII (1988), losing both to the San Francisco 49ers.