Son of former rallycross champion Franz Wurz, Alex took the BMX championship in 1988 before turning his attention to four wheels. After a couple of years in karting, the Austrian moved up through Formula Ford and the German F3 series.

By 1996, Wurz was busy in various categories of motorsport, the highlight being the Le Mans win in the Joest Porsche WSC95. Tests followed with Sauber and Benetton and his Formula One career was firmly on track.

Taking a testing role with Benetton Renault in 1997, Wurz got the call-up to the race seat when race regular Gerhard Berger was sidelined due to illness. He made his F1 debut in Montreal retiring with a gearbox problem and a spin followed in France. However, a strong performance at the British Grand Prix saw him take his first podium position as he chased team-mate Jean Alesi across the line.

With Berger back to full fitness, Wurz sat out the remainder of the season before signing up as race driver with the team. Wurz would finish eighth in the standings in his first full season with fourth places in Brazil, Argentina, Spain, Canada and Britain, and a fifth in France. His season was remembered mostly for his battle with Michael Schumacher at Monaco and his spectacular roll at the start of the Canadian Grand Prix.

While 1998 was a good season for Wurz, the following year was disappointing with two-points paying finishes. 2000 saw Wurz in the points just once at Imola and the Austrian was dropped from the line-up at the end of the season.

A new career with McLaren Mercedes followed as he took the testing option with the team. Rumours of a race-deal with Jaguar Racing came to nothing and Wurz busied himself behind the scenes with his test role. Five years on from his last points result at San Marino, McLaren drafted Wurz in to replace the injured Juan Pablo Montoya. A confident and aggressive race from Wurz showed that he had lost none of his pace displayed back in 1998 and he brought the McLaren home in third position to claim the second podium position of his career.

The race outing was a one-off at McLaren and Wurz opted to leave the team at the end of the season to take a testing role with Williams Cosworth.

The move paid off as Wurz teamed up alongside Nico Rosberg at Williams in 2007. Although Wurz took third position in a rather chaotic race in Montreal and a fourth place finish at Nurburgring, the season as a whole was a disappointment with Wurz struggling in qualifying in particular. Following the Chinese Grand Prix in which he finished 12th, the team promoted test driver Kazuki Nakajima to the race team and Wurz sat out the season finale in Brazil stating that he was to retire from active competition in Formula One.

For 2008, Wurz joins the Honda team as test driver.