He studied at the University of Budapest graduating with a BSc in 1928, continuing as a postgraduate, gaining a PhD.
He left Hungary in the years leading up to World War II and relocated to Scotland in the early 1930s.
In 1933 Koller obtained a DSc from the University of Edinburgh with a thesis on the dynamics of chromosome movement.
His proposers were Francis Albert Eley Crew, Alan William Greenwood, Sir Alick Buchanan-Smith, Baron Balerno, and James Nichol Pickard.
In 1946 he moved again, this time to the Chester Beatty Research Institute, first as Cytologist then as Professor of Cytogenetics.