Coptic Christians make up 10% of Egypt’s population of 80 million.
After attacks on Coptic Christians in recent years, Pope Shenouda urged officials to do more to address the community’s concerns.
Pope Shenouda led the church, one of the oldest Christian communities in the world, for four decades.
His political adviser Hany Aziz told Reuters news agency that Shenouda “died from complications in health and from old age”.
He had returned recently to Egypt after seeking treatment abroad.
Pope Shenouda was born Nazeer Gayed into a devout Christian family on 3 August 1923 in Asyut, Upper Egypt, and became a monk in 1954, taking the name Shenouda.
After Pope Cyril died in 1971, Shenouda was enthroned as Pope of Alexandria.
He fell out with President Anwar Sadat, who in 1981 sent him into internal exile. He was allowed back to Cairo by President Hosni Mubarak four years later.
The BBC’s Jon Leyne in Cairo says Pope Shenouda sought to protect his Christian community amid a Muslim population by striking a conservative tone and lending tacit to President Mubarak’s rule.
Whoever succeeds him now faces the task of reassuring the Coptic community as the Islamists of the Muslim Brotherhood look on the verge of sharing power in Egypt for the first time, our correspondent says.
Many younger Copts will now be looking for a leader who can help redefine their community’s role in a rapidly changing post-Mubarak Egypt, our correspondent adds.
The reign of Pope Shenouda III saw a huge expansion of the Coptic Orthodox Church beyond its traditional Egyptian base.
A passionate advocate of unity among the Christian churches, his sometimes radical ideas brought him into conflict with other Church leaders.
His reign also saw clashes with former Egyptian President Anwar Sadat, particularly over the future of his country’s relationship with Israel.
Pope Shenouda was born Nazeer Gayed into a devoutly Christian family on 3 August 1923 in Asyut, Upper Egypt.
He graduated with a history degree from Cairo University and began attending Coptic theological classes at night.
He became a monk in 1954, taking the name Father Antonios el-Syriani, and lived in a cave as a hermit for six years, spending his time praying and meditating.
In 1962 Pope Cyril IV ordained him as a bishop and he took the name Shenouda after the 4th century scholar Saint Shenouda the Archimandrite .
He was also made Dean of the Coptic Orthodox Theological Seminary eventually tripling the number of students studying at the centre.
Israel row
He achieved a reputation as a radical clashing with pope Cyril on a number of issues, including the right of congregations to select their own bishops.
Pope Cyril died in March 1971 and Shenouda was enthroned as Pope of Alexandria the following November, the first such ceremony in St Mark’s Coptic Cathedral in Cairo, which was inaugurated three years previously.
His first major test came in 1976 when the Patriarch of Ethiopia, Abuna Theophilos, was arrested by the Marxist regime in Addis Ababa and eventually executed.
Shenouda refused to recognise the replacement patriarch appointed by the Ethiopian government and relationships between the two churches were severed for 30 years.
He was exiled from Egypt by President Anwar Sadat in 1981 for refusing to hold public celebrations to mark Easter.
A number of other Church figures were arrested and Shenouda only returned to Cairo in 1985, three years after Sadat’s assassination.
He also fell out with Sadat because of Shenouda’s views over the relationship between Egypt and Israel.
The Pope banned members of his congregation from visiting Christian sites in Israel stating that Christians should only go to Jerusalem hand in hand with Muslims once the conflict between Israel and Arab states was resolved.
His pronouncement was attacked by Copts, particularly those outside Egypt, who accused their pope of mixing politics and religion.
Pope Shenouda’s reign saw a huge expansion in the Coptic Church particularly in the United States where the four churches in existence in 1971 have grown to more than 100.
There has been a similar growth in Australia and New Zealand as well as an increase in the number of churches in Europe and the UK.
His commitment to the concept of the unity of Christian churches took him to the Vatican in 1973 to meet Pope Paul VI, the first Coptic Pope for 1,500 years to go to Rome.
He believed that Christian unity was a matter of faith rather than of jurisdiction.
“Christian people, being fed up with divisions, are pushing their Church leaders to do something about church unity and I am sure that the Holy Spirit is inspiring us,” he told a forum in 1974.