The 64-year-old Bishop of Regensburg in Germany replaces the American Cardinal William Levada, who has reached the mandatory age for retirement.
The post is regarded as the third most important in the church and was occupied by Pope Benedict before his election in 2005.
Bishop Mueller is regarded as a generally orthodox theologian.
However, in 2004 he co-authored a book with his former teacher, Father Gustavo Gutierrez, the founder of liberation theology.
Pope Benedict is credited with having subdued the largely Latin American movement during his term as Prefect, believing it was too close to Marxism.
On the other hand, Bishop Mueller has been responsible for publishing Pope Benedict's writings.
In recent years, the Church has insisted that all allegations of clerical child sexual abuse be sent to the Prefect of the Congregation for the Doctrine of the Faith. He then decides whether they should be handled by his office or by the local church.
On his promotion, Bishop Mueller has been made an Archbishop.
Pope Benedict has strong links with Dr Mueller's old diocese.
As Professor Ratzinger he taught theology in the University of Regensburg before he was appointed Archbishop of Munich.
His brother, Father Georg Ratzinger still lives in the city and their parents and sister are buried there.