Zoran Stefanović had previously attempted to enter Formula One in 1996 and again in 1997, when he tried to acquire the remains of Lola's abortive entry that year. The team was built with the backing of Serbian company AMCO, which gave Stefan Grand Prix the rare capacity among applicants of the era to build its own chassis rather than outsourcing design to third parties.
When the FIA opened applications for new teams to join the 2010 grid, fifteen organisations applied, with three selected in July 2009: Campos Meta, Manor Grand Prix, and Team USF1. Stefan Grand Prix was rejected. Stefanović filed a complaint with the European Commission, arguing the selection process was biased in favour of teams using the new Cosworth engines and against constructor-class teams that could build their own cars. He cited Prodrive's similar rejection as corroborating evidence. He also secured the support of Mike Coughlan, the former McLaren engineer.
When Toyota withdrew from Formula One in November 2009 following a wave of manufacturer exits, Stefanović acquired the rights to use Toyota's chassis, gearbox, and 2010-specification engine, and hired several former Toyota team members. The FIA moved to fill the vacant thirteenth grid slot; after weighing Stefanović against Peter Sauber, the spot was awarded to Sauber.
Despite lacking an entry, the team shipped its equipment to Bahrain International Circuit ahead of the 2010 season opener and attempted to test its car at Autódromo Internacional do Algarve in February, but Bridgestone — the sole F1 tyre supplier — refused to provide race-specification tyres to a team without an official entry, forcing cancellation. The Toyota-designed car, designated the Stefan S-01 with the Stefan RG-01 engine, was fired up for the first time on 19 February 2010, liveried in Serbian national red. Kazuki Nakajima, whose career had been supported by Toyota, was confirmed as the first signed driver. Stefanović also hoped to sign 1997 world champion Jacques Villeneuve. Attempts to merge with the struggling Team USF1 were unsuccessful, and the partnership with Toyota Motorsport formally ended in March 2010 without the team having run in a race.
For 2011, the FIA re-opened the entry process following USF1's withdrawal, and Stefan Grand Prix was among fifteen organisations to apply. In the week before the 2010 Italian Grand Prix, the FIA announced that none of the prospective entrants met minimum funding or engineering requirements, leaving the grid slot vacant.
Stefan Grand Prix submitted another application when the FIA reopened the entry process for 2015, but Stefanović withdrew the application in February 2014. In July 2017 Stefanović announced plans for a 2019 entry, citing a facility in Parma and staff already assembled, though nothing came of the bid.
Stefan Grand Prix is remembered as one of the most persistent and contentious outsider efforts in modern Formula One history. The team's legal challenges to the entry selection process raised lasting questions about transparency in the FIA's expansion process. It remains the most prominent attempt to establish a Serbian presence in the sport's top tier, and its Toyota-derived hardware represented a genuine technical programme that never found a grid to compete on.