Named after Teddy Kollek, Jerusalem mayor from 1965-1993, this park is located in the valley just below Jaffa Gate. The highlight is a fountain with 256 water spouts which performs a choreographed water show once every 2 hours (check the schedule before coming); you are encouraged to run through the fountain on a hot day. There is also a constantly running water feature which is good for cooling off at any time. There's also some archaeology here, but it's unimportant by Jerusalem standards.
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A large water reservoir just outside the Old City. It likely dates to the time of Herod, and has been remodeled since then. Now, it is empty of water, and used to hold concerts and other large events.
The first neighborhood of Jerusalem to be built outside the Old City. The Mishkenot Sha'ananim housing project and the landmark windmill were finished in 1860; the rest of the neighborhood was built in the 1890s. Nowadays the neighborhood is quaint, beautiful, surrounded by parkland, and with a direct view of the Old City walls. It's a delightful neighborhood to walk around, both during the day and at night.
Between 1948 and 1967, West Jerusalem and Mount Zion were under Israeli control, but the only access between them was vulnerable to Jordanian sniper fire from the Old City walls. To avoid this fire, a secret tunnel was dug so that Mount Zion could be safely supplied. Part of the tunnel was neglected and destroyed after 1967, but the remainder has been restored and you can walk up the mountain through it.
A West Jerusalem landmark, built in 1933. There is a tower with views of the surroundings. The view is not bad, but less impressive than one might expect (you can see part of the Old City walls, and the top of Dome of the Rock, but little else of note). Going up the tower costs ₪20 - inquire at the main desk and you should be able to go up with little delay. There are reports that unaccompanied individuals are not allowed up, so go with someone else. Besides the tower, the whole building and its surroundings are architecturally interesting and worth walking around.
An interesting museum about the heritage of Jews from North African countries. You can only visit by advance reservation and as part of a group of 20 or more. Contact them and they might be able to arrange something.