Between 600 and 1650, what challenges did the Popes face to their rule, and why did their influence diminish after the sixteenth century?
The Papacy is a title which began to be claimed by the Bishops of Rome in the third and fourth centuries. Although many Christians in the Latin West agreed that the Bishop of Rome had a legitimate claim to the title of Pope and Vicar of Christ and authority over the entire church, many parts of the church considered the Bishop of Rome no more important than any of the other five leading bishops (the Pentarchy), having authority only over his own diocese, and, as is still the case among the Eastern Orthodox and many Protestant Christians, do not even use the term Pope to refer to the Bishop of Rome.
Next, in the Investiture Controversies, kings disputed the right of Popes to appoint bishops and abbots in their realms.
Finally, it diminished because many "heretical" movements such as the Lollards and the Albigensians, and eventually the Protestants, did not accept Papal authority.