British racer started karting at the age of eight before moving to Formula Ford with Martin Donnelly Racing. A move to Van Diemen followed in 2002 where he finished fourth in the championship.

Conway moved to Formula Renault in 2003 with Fortec, finishing fourth in the championship and then went on to take the title with the same team the following year.

Formula 3 was the next step, again with Fortec, where Conway would finish third in the championship in 2005. A move to RRR in 2006 gave Conway the prestigious F3 title and Macau Grand Prix victory.

Conway competed in the GP2 championship with Super Nova in 2007, finishing 14th in the championship. While the GP2 season did not go as well as planned, he joined the Honda Young Driver Programme and for 2008 he will test with the F1 team as well as racing in GP2 with Trident Racing.