The Minardi family had deep roots in motorsport. Giancarlo's grandfather ran a Fiat dealership in Faenza from 1927, and his father Giovanni built and raced his own cars in the late 1940s. Giancarlo took over the family racing operation in the early 1970s, managing the Scuderia del Passatore in Italian Formula competitions. The team, sometimes renamed Scuderia Everest for sponsorship reasons, ran in Formula Two from 1975 to 1979, fielding drivers including Elio De Angelis and Clay Regazzoni. With financial backing from patron Piero Mancini, Giancarlo established the Minardi racing team as a proper Formula Two constructor from 1979, winning the 1981 Misano round with Michele Alboreto before graduating to Formula One for 1985.
Minardi entered Formula One in 1985 with a single car designed by Giacomo Caliri, powered initially by a Ford Cosworth DFY before the intended Motori Moderni turbo engine became available. Pierluigi Martini drove for the team that year, scoring no points but laying the foundation for what would become a long association between driver and constructor.
The team expanded to two cars in 1986 and gradually built a presence in the midfield through the late 1980s and early 1990s. Martini scored the team's first championship point at the 1988 United States Grand Prix. In 1989 Minardi became the lead entrant for Pirelli's return to Formula One. The 1990 United States Grand Prix brought the team's only front-row start when Martini qualified second, aided by Pirelli's qualifying compound. That same year he led a lap of the Portuguese Grand Prix, the only occasion a Minardi car led a World Championship race. In 1991 Minardi became the first team to run customer Ferrari V12 engines in the modern era, and the car's best results came that year: Martini finished fourth at both San Marino and Portugal, lifting the team to seventh in the Constructors' Championship, their highest-ever placing. In 1992 the team switched to Lamborghini V12 power. Christian Fittipaldi's fourth place in the 1993 South African Grand Prix stands as one of the team's strongest single results.
By the mid-1990s the team's finances deteriorated as the smaller constructors struggled against manufacturer money. In 1994 Minardi merged with BMS Scuderia Italia. Flavio Briatore briefly took a stake in 1995, and businessman Gabriele Rumi became co-owner and chairman from 1996. When Rumi's ill health forced his withdrawal at the end of 2000, the team was on the verge of collapse.
Australian businessman Paul Stoddart purchased Minardi in early 2001, merging it with his European Racing Formula 3000 operation. The team's most celebrated act in this period was giving Fernando Alonso his Formula One debut in 2001 at the age of 19. Though Alonso did not score points that year, Renault immediately signed him for 2002. His replacement, Mark Webber, finished fifth on debut in Melbourne. Stoddart became a vocal campaigner within the paddock for cost reduction and independent team rights, frequently clashing with FIA president Max Mosley. In 2004 Hungarian driver Zsolt Baumgartner scored the team's first championship point in over two years with eighth place at the United States Grand Prix.
In 2005 Red Bull GmbH purchased the team, renaming it Scuderia Toro Rosso for 2006. Increased funding and Red Bull technology transformed the operation; Sebastian Vettel won the 2008 Italian Grand Prix with Toro Rosso, and the team later claimed victory again as Scuderia AlphaTauri at Monza in 2020 with Pierre Gasly. In 2024 the team was rebranded as Visa Cash App RB.
Over 21 seasons Minardi scored 38 championship points across 340 starts and entered 37 drivers. Pierluigi Martini drove 103 Grands Prix for the team, the most of any Minardi driver. Alumni include two-time World Champion Fernando Alonso, race winners Alessandro Nannini, Giancarlo Fisichella, Jarno Trulli, and Mark Webber, as well as Le Mans overall winners Michele Alboreto, Martini, and Marc Gené.
Within Formula One circles Minardi was widely respected for its lack of corporate culture, its resistance to employing pure pay drivers, and the quality of engineering it achieved relative to its always-limited resources. Giancarlo Minardi re-acquired certain name rights from 2006 and continued involvement in junior categories, while Stoddart used the Minardi name briefly in Champ Car with Minardi Team USA. The Faenza factory itself endured, eventually becoming the headquarters for successive Red Bull junior teams.