Born in Leipzig, Germany, he was educated at the Albert Gymnasium in that city from 1921 to 1929 and went on to study medicine at the University of Leipzig. He graduated in 1934 and fled to Britain in February 1935, the rise of Hitler having made his mixed Russian-Jewish heritage dangerous. He went to work at UCL, initially under the tutelage of Archibald Vivian Hill. He finished his PhD in 1938 and won a Carnegie Fellowship to study with John C. Eccles at Sydney Hospital. He was naturalised in 1941 and joined the Royal Australian Air Force in 1942. He spent the war in the Pacific as a radar officer. He married Marguerite Penly in 1945 and returned to UCL as an assistant director in 1946. Back in England he also worked with the 1963 Nobel prize winners Alan Hodgkin and Andrew Huxley. Katz was made a professor at UCL in 1952 and head of biophysics, he was also elected to the Royal Society. He stayed as head of biophysics until 1978 when he became emeritus professor.
His research uncovered fundamental properties of synapses, the junctions across which nerve cells signal to each other and to other types of cells. By the 1950s, he was studying the biochemistry and action of acetylcholine, a signalling molecule with which synapses linking "motor nerves" to muscles stimulate contraction. Katz won the Nobel for his discovery that neurotransmitter release at synapses is "quantal"--that is, that at any particular synapse the amount of neurotransmitter released is never less than a certain amount, and if more is always an integral number times this amount. This circumstance arises, scientists now know, because, prior to their release into the synaptic gap, transmitter molecules reside in like-sized subcellular packages known as synaptic vesicles (more at exocytosis).
Katz's work had immediate influence on the study of organophosphates and organochlorines, the basis of new post-war study for nerve agents and pesticides, as he determined that the complex enzyme cycle was easily disrupted. He went on to study the biochemistry of the pineal gland, especially the production of melatonin (discovered in 1948).