Abraham Isaac Katsh was born in Poland on August 10, 1908, and immigrated to New York in 1925. He completed his early education in Poland and Palestine before receiving a number of degrees from New York University (NYU). Katsh received a bachelor of science from Washington Square College (1931), a master of arts from the School of Education (1932) and a doctor of jurisprudence from the School of Law (1936), all at New York University. He later completed graduate work at the Islamic Institute of Princeton University, then went on to receive a Ph.D. from Dropsie College for Hebrew and Cognate Learning in Philadelphia. In 1964 Katsh received an honorary doctorate of humane letters from Hebrew Union College.

In 1933 Katsh joined the faculty of NYU and introduced the first course in modern Hebrew to be taught at an American university. He served as a lecturer, Professor of Hebrew Culture and Education, Professor of Hebrew and Near Eastern Studies, and Distinguished Professor of Research and Director of the Hebrew Institute at NYU. Among Katsh's other accomplishments at NYU were the founding of the Jewish Culture Foundation (1937) and the establishment of the NYU Library of Judaica and Hebraica. Between 1947 and 1967, Katsh also developed and directed the American Israel Student and Professorial Workshop, a summer workshop conducted in Israel and co-sponsored by the State Department and Federal Office of Education. In recognition of his contributions to the university, the chairmanship of the Hebrew Studies Department at NYU was endowed with Katsh's name in 1957. In 1967 Katsh left NYU to become the president of Dropsie College, a position he held until 1976.

Katsh was the author of numerous books, including many on Hebrew language and literature, as well as Jewish history and culture. One of his first major successes was a translation of Einstein's General Principle of Relativity into Hebrew from its original German. He later published a book entitled Judaism in Islam (1954) that explored Talmudic and Biblical influences in the Koran. In 1956 Katsh visited the Soviet Union, Poland and Hungary, where he discovered a wealth of Judaica manuscripts and documents that had been concealed from the western world since the Russian Revolution in 1917. On this trip and five successive ones, Katsh persuaded the Soviet authorities to allow him to microfilm these rare documents, including thousands of fragments from the Cairo Genizah, and to share his findings with the world. Katsh published these microfilmed images in a number of volumes, including Catalogue of Hebrew Manuscripts Preserved… (On Microfilm) in the U.S.S.R. (2 vols., 1957–58), Ginze Mishna (1971) and Ginze Talmud Bavli (1975, 1979). In 1965 Katsh published the translation of a previously undiscovered diary from the Warsaw Ghetto under the title Scroll of Agony: The Warsaw Diary of Chaim A. Kaplan. Aside from his numerous books, Katsh was also a prolific author in both Hebrew and English newspapers and academic journals.

Katsh served as a visiting scholar in the Oxford Centre for Post Graduate Hebrew Studies in 1977-78 and was also a visiting scholar at Mishkenot Sha'ananim in Jerusalem in 1978. An annual scholarship was established in his name in 1980 at the Hebrew University in Jerusalem by his former students and friends from his time at NYU.

Katsh passed away in 1998.