He has served in the position for nearly 14 years and is the longest serving archbishop in the worldwide Anglican Communion.
The clergyman is retiring on his 70th birthday, just months after the death of his wife from cancer.
In September 2016 he came out in support of gay marriage, questioning why Christians would want to deny same-sex couples the right to marry.
He said studying the Bible in its full context had led to a very different view of same-sex relationships than that traditionally held by the Church.
Currently same-sex weddings are not allowed in the Church in Wales but this policy could change in the next few years.
Dr Morgan has also overseen a change in Church law to enable women to be ordained as bishops and the implementation of a strategy called 2020 Vision, aimed to help the church grow and prosper in the approach to its centenary year.
Originally from the small mining village of Gwaun-Cae-Gurwen in the Swansea valley, Dr Morgan was elected as the 12th Archbishop of Wales in 2003, following Dr Rowan Williams after his appointment as Archbishop of Canterbury.
On Sunday more than 500 people attended a farewell service at Llandaff Cathedral.
Speaking previously Dr Morgan said: "It's been a rollercoaster ride but all along I have been sustained and inspired by the people I meet, day in day out, who live out God's love in every part of Wales through their commitment and devotion to their churches and communities.
"I would like to thank all those who have supported, shared and upheld me in my ministry over the years, particularly since Hilary's death - the loss of her love, encouragement and friendship has been enormously hard to bear."
