Graif received a Doctor of Medicine degree from the University of Bologna in 1973.
Graif began his career as a resident at Rambam Medical Center in 1975, where he trained in radiology. Five years later he took a position of a staff radiologist at Chaim Sheba Medical Center. Since 1989, Moshe was appointed a chairman of the department of medical imaging at Tel Aviv Sourasky Medical Center. In 1984, he joined London Hammersmith Hospital for a year as a research fellow at the MRI (Magnetic Rsonance Imaging) unit, with a view to bringing the new technology and clinical practice to Israel.
Also since 1992 he was a professor of medical imaging at the Faculty of Medicine of Tel Aviv University. Graif was a visiting professor at the MRI division of Thomas Jefferson University Hospital in 1995-1996 and an adjunct associate professor of radiology at the same hospital in 2000.
In addition, Moshe served as a visiting Professor at Saint Paul’s Hospital in 2007 and at the Italian Hospital of Buenos Aires University in 2010. He was MSK area editor of the Journal of Clinical Ultrasound since 1995 and a reviewer for many publications, including the European Journal of Radiology and the Israel Journal of Medical Sciences.
Graif was a chairman of the Israel Radiological Association in 1999-2011. He was a member of the Executive Committee of the Scientific Council of the Israel Medical Association, the National Council of Medical Imaging for the Israeli Ministry of Health and Chief Radiology Counsellor for the Maccabi Health Services Organization.
He served as co-president of the French-Israeli Association of Medical Imaging since 2002 and the East Mediterranean Congress of Magnetic Resonance from 2002 to 2006. Moshe was an honorary member of the French Radiological Society.
On August 7, 1980 Moshe Graif married Yael Green. They have 3 children.