Jesper received his PhD from Aarhus University in Denmark (1993), based on work on DNA topoisomerases. He was then a postdoc in Roger Kornberg's laboratory at Stanford University (1993-1996), before starting his lab at Clare Hall Laboratories (1996). He received tenure in 2000. His lab became part of the Cancer Research UK London Research Institute in 2002 and is now part of the Francis Crick Institute.
In recognition of his laboratory's contributions to scientific progress, Jesper was elected to the European Molecular Biology Organization (EMBO) in 2003, to the Royal Society in 2009 and the Academy of Medical Sciences in 2018.