The 270-foot-diameter (82 m) Alexander Coliseum opened in 1956 at the intersection of 10th Street and Fowler on the northeast end of the Georgia Tech campus. The building was named for William A. Alexander, Georgia Tech's football coach from 1920 to 1944 and the third athletic director (after John W. Heisman); his tenure as coach included the 1928 Rose Bowl team (the 1928 season, the Rose Bowl itself was played on January 1, 1929). The huge dome was constructed by Calvert Iron Works, Co., a local Atlanta company that built many Atlanta landmarks. It was designed and built without any support pillars to obstruct the view of any fans during games. It served as a replacement for the Third Street Gymnasium (later known as the Heisman Gym) on the campus of Georgia Tech, an 1,800 seat arena opened just 18 years before. The arena's original capacity was 6,996 seats, though crowds larger than that sometimes assembled for big games.

Between 1956 and 1996, the Coliseum had undergone three major renovations. In 1986, 2,150 seats were added in what was previously an upper walkway around the rim of the arena. In 1989–1990, 750 seats were added in the end zone areas. The last major renovation was during 1995–1996, prior to the 1996 Summer Olympics. The floor was lowered 4 feet (1.2 m) to increase seating, 12 luxury suites were added, and many of the benches were replaced with chairback seats. Sightlines were also improved for those sitting in the first few rows of the side of the court opposite the benches. Connected to the south end of the Coliseum are the Luck Building, and the Coliseum Annex. The facility was renamed Alexander Memorial Coliseum at McDonald's Center for 10 seasons from 1996 to 2005 in conjunction with a $5.5 million donation to help pay for the mid-1990s renovation, which increased seating capacity to 9,191.

The Coliseum played host to the Atlanta Hawks after they moved from St. Louis while The Omni was under construction, and again between 1997 and 1999, after the Omni was demolished and while Philips Arena was being built on its site. During the latter period, the Hawks played most of their home games at the Georgia Dome with the remainder at the Coliseum. The team again played at McCamish Pavilion during October 2017 for preseason games, when Philips Arena improvements forced the Hawks to play there a third time.

During the 1996 Summer Olympics, the arena hosted the Olympic boxing tournament.

For most of its life, the Coliseum has hosted many rounds of the Georgia High School Association men's and women's state tournament games. The first integrated high school state tournament in Georgia history was played there before record overflow crowds in 1967.

The arena received its nickname, "The Thrillerdome," from former Tech color radio announcer and former ESPN broadcaster Brad Nessler, for the many close games it witnessed during the 1983-84 season.

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