Born in 1921, Epstein studied medicine at Trinity College, Cambridge and the Middlesex Hospital Medical School. He worked at Middlesex Hospital and in 1961 attended a lecture given by visiting physician Dr Denis Burkitt about children with exceptionally large facial tumours in Uganda. These tumours are now understood to be Burkitt lymphoma. Epstein had previously been working with viruses that caused cancer in chickens, and suspected that a virus might also be involved in this new case. He asked Burkitt to send him tumour samples, determined to be the first to discover a cancer-causing virus in humans.

These tumours would travel in culture fluid from Kampala to London for Epstein to analyse, but initially a virus could not be detected. However, as a result of fog on one journey, the flight path from Kampala to London was diverted to Manchester. This longer period of travel resulted in the sample being cloudy by the time it reached Epstein in London.

Alongside his colleagues Yvonne Barr and Bert Achong, Epstein used an electron microscope to look at the culture of these cells and discovered virus particles within it that resembled one of the family of herpes viruses that was undescribed at the time. In March 1964 this discovery was published in The Lancet medical journal and by 1965 the virus was named after two of the three scientists that discovered it: the Epstein-Barr virus (EBV). Approximately 90-95% of adults worldwide are infected with EBV. However, EBV-associated cancers are uncommon, and most people infected with EBV will not develop cancer.

The discovery of the viral cause of Burkitt lymphoma has led to further research into viruses associated with cancer.