Robert started karting in 2004, winning championships across Europe but mostly Italy.
He moved to single seaters racing in Italian F4 Championship with Cram Motorsport, finishing 16th in the championship after driving 6 races. In the following year, Robert moved to Mücke Motorsport to compete in the F4 championship, behind Prema’s duo of Guanyu Zhou and Ralf Aron. At the same time he was also competing in ADAC Formula 4 championship with ADAC Berlin-Brandenburg e.V., finishing 4th.
By 2016, he moved to Formula Renault 2.0 with Josef Kaufmann Racing whilst claiming two victories in the Northern European Cup and finished 6th in the standings. In the Eurocup, Robert finished 8th. In the following year, he raced with R-ace GP, finishing 3rd in the standings.
Shwartzman made his Toyota Racing Series debut with M2 Motorsport. He finished all fifteen races in the top-five and was the only driver in the season to do so. He won the Denny Hulme Memorial Trophy and the series title, classifying ahead of both of his teammates Richard Verschoor and Marcus Armstrong.
Robert made his Formula 3 debut in the FIA Formula 3 European Championship with Prema Powerteam, driving alongside Guanyu Zhou, Ralf Aron, Marcus Armstrong and Mick Schumacher. He finished 3rd in the final standings behind Dan Ticktum and won the rookie championship. He returned to Formula 3, now named FIA Formula 3, again with Prema Powerteam, lining up beside Marcus Armstrong and Jehan Daruvala. He won 3 times and was on the podium 10 times, meaning he won the championship in Sochi feature race.
During both Formula 3 seasons, he competed in Macau Grand Prix, both with Prema Racing. In the qualifying race of 2018, he finished 10th, and in the main race, he finished 9th. In the following year he finished 2nd in the qualifying race and was hit in the main race.
Robert joined Prema Racing for the 2020 campaign alongside Mick Schumacher. [insert details here x]
For 2021, he was announced to be retained within the team, but this time joined with Formula 3 champion Oscar Piastri. He would be driving with the #1 car.