Eddy began his research career at the University of Leeds working with Ken Powell. After postdoctoral research at McMaster University in Canada and the Paterson Institute for Cancer Research in Manchester he joined the Wellcome Research Laboratories as Head of Herpes Virus Research. At Wellcome he became Head of the Gene Targets Group and following the acquistion of Wellcome by Glaxo he became Head of Antiviral Research at GlaxoWellcome.
Eddy left GSK in 2000 to join Medivir as Senior Director of Lead Discovery and Head of their UK operations. Whilst at Medivir he played a significant role in several deals, including the outlicensing of the HCV protease inhibitor programme to Tibotec. The total value of deals converted at Medivir was $360M. Eddy became the Chief Executive Officer of Domainex in May 2008. Eddy is a member of the SAB of Tibotec and a member of the Infection and Immunity Board and the Translational Research Group at the MRC.