As reported in Wikipedia:

The creation of Fiesta Bowl grew out of the frustration of the Western Athletic Conference in trying to obtain suitable bowl invitiations for its champions. In 1968 and 1969, respectively, champions Wyoming and Arizona State failed to obtain any invitation, while in 1970, undefeated Arizona State was bypassed by the major bowls and had to settle for a less prestigious Peach Bowl appearance. The Fiesta Bowl therefore initially provided an automatic bowl tie-in for the Western Athletic Conference champion.

The 1971 inaugural game featured another top ten Arizona State squad against top twenty opponent Florida State. By 1975, the game was able to attract Big Eight co-champion Nebraska to play undefeated Arizona State in a matchup of top five teams. In 1977, the game was again able to attract a top five opponent in Penn State.

In 1978, Arizona and Arizona State each joined the Pac 10 Conference, and the Fiesta Bowl’s tie-in with the Western Athletic Conference ended. The game continued to attract high quality matchups, however, and beginning with the 1981 game shifted to New Year’s Day with the other major bowl games.

A major breakthrough occurred in 1987 when the top two teams in the country, Miami and Penn State, agreed to play for the National Championship in the Fiesta Bowl. Both universities were independents at that time, all of the so-called major bowls had tie-ins with conferences, and of the other bowls only the Fiesta Bowl had the resources to pull off a matchup of this magnitude. The 1987 game, won by Penn State 14-10, was the most watched college football game, in terms of television viewership, until the 2006 Rose Bowl Game. Two years later, #1 Notre Dame played undefeated#3 West Virginia for the National Championship at the Fiesta Bowl. By this time, the Fiesta Bowl’s ability to stage games with national championship implications propelled it to major-bowl status in the minds of college football fans, replacing the Cotton Bowl as a member of the postseason’s “top four.”

For the 1991 game, several major universities declined invitations due to the State of Arizona’s decision at that time not to adopt the Martin Luther King Holiday.

In 1992, the Fiesta Bowl was invited to participate in the Bowl Coalition, a predecessor to the Bowl Championship Series, assuring the game would feature major conference champions or prestigious runners up, and the possibility of hosting further national title games. When this series incorporated a title game with a preset rotating site beginning in 1996, the Fiesta Bowl was the first to host the game, featuring undefeated #1 Nebraska playing undefeated#2 Florida for the National Championship. Finally, with the addition of the Big Ten and Pac 10 conferences to the rechristened Bowl Championship Series, the Fiesta Bowl became a permanent fixture in the four year BCS National Championship Game rotation. In 1998, the Fiesta Bowl featured the first BCS National Championship Game, which Tennessee won over Florida State, 23 to 16.