Wikipedia
Fargo opens with the following text:

THIS IS A TRUE STORY. The events depicted in this film took place in Minnesota in 1987. At the request of the survivors, the names have been changed. Out of respect for the dead, the rest has been told exactly as it occurred.

Although the film itself is completely fictional, the Coen brothers claim that many of the events that take place in the movie were actually based on true events from other cases that they threw together to make one story. Joel Coen said, “We weren't interested in that kind of fidelity. The basic events are the same as in the real case, but the characterizations are fully imagined.” He later noted, “If an audience believes that something's based on a real event, it gives you permission to do things they might otherwise not accept.”