The Pope -- Bishop of Rome. While his concerns are mainly with the Universal Church, the Pope remains of course the Bishop of Rome, which is the central reason of why he is the Pastor of the Universal Church in the first place.
But since for many centuries, they have been so busy with the Church outside the City, they have for many centuries kept a Cardinal-Vicar, who administer the diocese.
Also for many decades, especially since Rome became once again a bustling modern metropolis, the central area of the City (which includes most of what the ancient, the modern, and the pilgrims and tourists identify as Rome proper) has been administered as a whole: the map below, from the pontificate of Benedict XVI, identifies this area, the Settore Centro, in red (the Vatican itself appears in yellow). The exact limits at the time had been defined by John Paul II.