De Angelis was born in Rome to a prosperous family; his father Giulio was a real estate developer and a multiple world champion in offshore powerboat racing. After a brief spell in karting, de Angelis won the Italian Formula Three Championship in 1977. The following year he raced in Formula Two with Minardi and the ICI British F2 Team, and also triumphed in the prestigious Monaco Formula Three race. His talent was noticed early: at the close of the 1977 season, the 19-year-old was on Enzo Ferrari's shortlist to replace Niki Lauda, and he tested the Ferrari at Fiorano. Ferrari ultimately chose Gilles Villeneuve, but de Angelis's ability was beyond doubt.
De Angelis made his Formula One debut in 1979 with Shadow, finishing seventh on his first start in Argentina and collecting three championship points by season's end. His pace caught the eye of Lotus boss Colin Chapman, who signed him to partner Mario Andretti from 1980. At 21 de Angelis became the youngest Grand Prix podium finisher of his era when he took second at the Brazilian Grand Prix at Interlagos.
His first victory arrived at the 1982 Austrian Grand Prix at the Österreichring, beating the Williams of Keke Rosberg by just 0.05 seconds. The win was the last occasion Colin Chapman celebrated by hurling his cap into the air; Chapman died in December 1982. In 1983 Lotus switched from the Cosworth DFV to Renault turbo engines, with mixed results, but 1984 brought a marked improvement. De Angelis scored 34 points and finished third in the World Drivers' Championship, the only driver in the top five that year not to win a race, demonstrating a level of consistency that set him apart.
In 1985 he was joined at Lotus by Ayrton Senna, recruited from Toleman. De Angelis secured his second and final victory at the San Marino Grand Prix after Alain Prost was disqualified post-race, and also claimed his last pole position in Canada. He finished fifth in the championship with 33 points, five behind his new teammate. Frustrated that the team's development efforts were increasingly directed toward Senna, he chose to leave Lotus at the end of the year.
For 1986 de Angelis moved to Brabham, replacing two-time world champion Nelson Piquet, with Riccardo Patrese as his teammate. The season was a difficult one. The BT55, designed by Gordon Murray, featured a low-line chassis intended to reduce drag and improve downforce, but the BMW four-cylinder turbo engine had to be tilted to 72 degrees to fit the package. This caused severe oil surge and compounded the engine's already poor throttle response. Brabham fell well off the pace of the leading teams, and de Angelis had yet to score a championship point by the time of his fatal accident.
On 14 May 1986, three days after retiring from the Monaco Grand Prix, de Angelis was testing the BT55 at Paul Ricard in France. Around midday the rear wing detached at speed, sending the car into a cartwheel over a crash barrier where it caught fire. The impact left de Angelis with only a broken collarbone and minor burns, but he was unable to free himself unaided from the wreckage. A shortage of marshals on the circuit delayed assistance, and it was 30 minutes before a helicopter arrived. De Angelis died 29 hours later in a Marseille hospital from smoke inhalation.
The accident brought Formula One's use of the full 5.812 km Paul Ricard layout to an end. Subsequent events at the circuit used a shortened configuration that bypassed the Verriere curves where the crash occurred and reduced the length of the Mistral Straight. The full layout did not host an F1 race again until 2018, when further modifications had been made. De Angelis's place at Brabham was taken by Derek Warwick, reputedly because Warwick was the only senior driver who did not immediately call team owner Bernie Ecclestone to request the vacant seat. Keke Rosberg, a close personal friend of de Angelis, retired from Formula One at the end of that season.
De Angelis was the last driver to die in a Formula One car until Roland Ratzenberger perished during qualifying at Imola in 1994, the day before Ayrton Senna was also killed there.
Beyond his racing ability, de Angelis was a concert-standard pianist. During the Grand Prix Drivers Association strike at the 1982 South African Grand Prix, when drivers locked themselves into a Johannesburg hotel to protest new Super Licence conditions imposed by the FIA, de Angelis entertained his colleagues with extended performances of Chopin and Mozart. This blend of artistic refinement and competitive brilliance made him a unique figure in the sport, the phrase "last gentleman player" attaching itself to him after his death.
Jean Alesi, who broke into Formula One in 1989, wore a helmet modelled on de Angelis's design in tribute. In 2017, de Angelis was honoured at the Ludovico Scarfiotti Memorial in Rome. He was buried in the Verano Monumental Cemetery in Rome.