Apeloig was born in Bukhara, Uzbekistan after his family fled from the Nazis after the invasion of Poland in September 1939. In 1947, when he was three years old, the family immigrated to Israel. He served in the Nahal Brigade and the paratroopers between 1962 and 1964. He studied chemistry and physics at the Hebrew University of Jerusalem, and completed his undergraduate (receiving his BA in physics and chemistry in 1967, and his masters in 1969) and graduate education there, including a Ph.D. in chemistry in 1974. He conducted postdoctoral research at Princeton University with Paul v. R. Schleyer and collaborated with Nobel Laureate John A. Pople. Apeloig joined the faculty of the Technion in 1976 and in 1983 he was appointed professor. He became the dean of the Faculty of Chemistry in 1995 until 2001 when he became the president of the Technion, replacing Amos Lapidot. In 2009 he was followed as President by Peretz Lavie.