Acosta participated in various national competitions, winning in 2017 in the PreMoto3 category. In 2018 he entered the Junior Moto3 World Championship. In 2019 he also raced in the Red Bull MotoGP Rookies Cup, finishing second and winning three races. He finished as champion the following year, winning six races in total, all six coming consecutively at the first six races of the season.
In 2021, Acosta made his debut racing in Moto3 class with the Red Bull KTM Ajo team. In his first race in Qatar, he finished in second place. On 4 April 2021, he won the Doha Motorcycle Grand Prix starting from pitlane, becoming the first rider in the history of Moto3 to accomplish this feat. Acosta would later win the next two races in Portugal and Spain, becoming the first rider in Grand Prix history to get on the podium in all of his first four races. He would win three more races during the season, in Germany, Styria, and Algarve, becoming champion in Algarve. He won the championship by 43 points, and became the first rookie champion of the Moto3 class since 1990, when Loris Capirossi won the 125cc title. Acosta was only one day older (17 years, 166 days) than Capirossi (17 years, 165 days) when they won their respective titles.
Acosta was promoted to Moto2 with Red Bull KTM Ajo in 2022, joining compatriot Augusto Fernández. He recorded his first Moto2 win during the Italian Grand Prix in Mugello. After missing two rounds due to a broken femur, he won another race in Aragón. Acosta won the 2023 Moto2 World Championship.
Acosta was promoted to MotoGP for 2024 with Red Bull GasGas Tech3. He finished eighth in the sprint and ninth in the race in Qatar. In Portimao, he achieved a podium in the main race, having overtaken experienced riders Brad Binder, Marc Marquez, and Francesco Bagnaia. At the Circuit of the Americas, he fought Marquez again, as well as Jorge Martin only to finish fourth in the sprint, but in the main race he led a race for the first time and finished second behind Maverick Vinales. He finished third in the sprint race and the main race of the Aragon GP. Acosta also scored his maiden premier class pole position at the Japanese Grand Prix. He finished sixth overall in his first MotoGP season, only two points behind factory rider Brad Binder.
Ahead of the Italian Grand Prix in Mugello, it was announced that Acosta had signed a multi-year deal with Red Bull KTM Factory Racing for the 2025 season. He scored five podiums to rank fourth in the championship. Acosta remained with the factory KTM team in 2026. He won the opening sprint race of the season in Thailand, and finished second in the main race, becoming the first KTM rider ever to lead the MotoGP World Championship.
Acosta holds several records in Grand Prix motorcycle racing. He is the youngest rider to have set a fastest lap in MotoGP, aged 19 years and 290 days. He is also the youngest rider to score a podium finish in MotoGP, aged 19 years and 304 days. Acosta is the youngest-ever rider to win a MotoGP Sprint, aged 21 years and 279 days, and the first KTM rider ever to lead the World Championship. In Moto2, he is the youngest world champion aged 19 years and 172 days (2023) and the youngest race winner aged 18 years and 4 days. He holds the record for most podiums in a single Moto2 season with 14 (2023), shared with Marc Márquez, Esteve Rabat, and Johann Zarco. He also shares the record for most fastest laps in a single Moto2 season with 8 (2023) with Franco Morbidelli. In Moto3, he is the youngest world champion aged 17 years and 166 days (2021).
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