Born in 1930 in Lydda, Palestine, Shammout and his family were amongst the 25,000 residents of Lydda expelled from their homes by Israeli soldiers. The Shammout family moved to the Gaza refugee camp of Khan-Younes. In 1950 Shammout went to Cairo and enrolled in the College of Fine Arts. After returning to Gaza in 1953, he held his first exhibition, which was a success. Shammout and Palestinian artist Tamam al Akhal participated in the Palestine Exhibition of 1954 in Cairo. The exhibition was inaugurated by then Egyptian President Gamal Abdul Nasser. Later in 1954, he moved to Italy and enrolled at the Academia De Belle Arti in Rome. He married al Akhal in 1959. Their work has been exhibited in several countries. Shammout became a part of the Palestine Liberation Organization (PLO) as the Director of Arts and National Culture in 1965. He also held the position of Secretary General of the Union of Palestinian Artists. He became Secretary General of the Union of Arab artists in 1969. In 1992 he and his wife, al Akhal, moved to Germany due to the Gulf War. After Germany, they settled in Jordan. He and al Akhal, returned to Lydda in 1997, facing the reality that their hometown was now part of Israel. He died on July 1, 2006 at the age of seventy-six.