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To get around the New City Sundays to Fridays: there is a good bus service with two main hubs: the Central Bus station in north Jerusalem and the Railway station in south Jerusalem. On Saturdays (the Jewish Sabbath) Israeli buses do not run. However if you are in the Old City region there are Arab Palestinian bus lines that serve nearby Arab villages.
Taxis can be hailed in the streets or booked by phone. For tourists there is a round tour bus (number 99) which as in many other capitals allows one to hop on and off as it goes around the city.
Inside the walls of the Old City there is only one bus line that uses small vehicles to navigate the narrow streets from the Jaffa Gate to the Western Wall(A major Jewish Holy Site). Otherwise in the Old City one has to walk.
With the notorious traffic jams in Jerusalem it is often easier to walk if you are not going to the suburbs.