Josephson was born in Cardiff and educated at Cambridge University, where he obtained his PhD in 1964. He remained at Cambridge and in 1974 was appointed to a professorship of physics.

His name is associated with the Josephson effects described in 1962 while he was still a graduate student. The work came out of theoretical speculations on electrons in semiconductors involving the exchange of electrons between two superconducting regions separated by a thin insulating layer (a Josephson junction). He showed theoretically that a current can flow across the junction in the absence of an applied voltage. Furthermore, a small direct voltage across the junction produces an alternating current with a frequency that is inversely proportional to the voltage. The effects have been verified experimentally, thus supporting the BCS theory of superconductivity of John Bardeen and his colleagues. They have been used in making accurate physical measurements and in measuring weak magnetic fields. Josephson junctions can also be used as very fast switching devices in computers. For this work Josephson shared the 1973 Nobel Prize for physics with Leo Esaki and Ivar Giaever.

Later, Josephson turned his attention to the study of the mind and has argued strongly for a connection between parapsychology and quantum mechanics.