Israel Finkelstein has written a very good article about the archaeological situation in the City of David.  I do not share Finkelstein’s view of the tenth-century BC poverty of the city, but with regard to modern political realities, he speaks much truth.  From Forward:

Confusion regarding this area begins with its name. Palestinians call it Silwan, but this is base propaganda aimed at the uninformed and uncritical international media. The Palestinian village of Silwan is located not in the City of David but rather to the east, on the other side of the deep Kidron Valley. Old photographs taken before the middle of the 20th century show the ridge cropping out south of the Temple Mount to be devoid of almost any buildings.
Jews and researchers of all backgrounds call the site the City of David — a name given to the ridge by early European explorers. Scholars agree that together with the Temple Mount and the southwestern part of the Old City, this ridge is the location of biblical Jerusalem.
[…]
This site should be revered as one of humanity’s great landmarks. Were it not for the political controversy surrounding the site, it would doubtless be high on the list of world heritage sites.
Allegations are sometimes heard in the media that work in the City of David is unlawful and not executed to the standards of modern archaeology. This is untrue. Fieldwork there is carried out according to law and — taking into account the difficulties of excavating in a built-up area — using sound field methods. All excavation projects are directed by seasoned archaeologists and inspected by the Israel Antiquities Authority.
[…]
Further to the east, the village of Silwan is built over unique, monumental Judahite rock-cut tombs from the 8th and 7th centuries BCE. Two of these tombs had ancient Hebrew inscriptions on their façades. But the tombs are neglected, flooded with sewage and filled with village garbage. And, of course, the greatest devastation to have recently been inflicted on Jerusalem’s archaeological heritage was the large-scale bulldozing a few years ago of buried antiquities on the Temple Mount by the Waqf, which administers the Islamic holy sites, in preparation for the construction of a massive underground mosque.

There is too much of value to excerpt, and I commend the entire article to you.

This photo is from the Jerusalem volume of The American Colony and Eric Matson Collection (Library of Congress, LC-matpc-05424).