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A great way to spend a weekend afternoon is in the DUMBO area of Brooklyn. The name stands for Down Under the Manhattan/Brooklyn Overpass. It's the area between the Manhattan and Brookyn bridges.
Like most cities, NYC is very good at revitalizing spaces that seem useless. In between the cobble-stoned streets and alleys are boutiques, famed pizzerias, ice cream shops, rock climbing and even a "beach."
If you're in Manhattan, you walk across the Brooklyn Bridge.
Facing the water, the Brooklyn Br. will be to the left and the Manhattan will be to the right (the third bridge to the BMW trio is the Williamsburg, which is further uptown, to the right).
Stroll along the waterfront taking in the small green spaces, the rocks along the shore, Pebble Beach, the playgrounds and the various vendors. Under the Manhattan bridge may be a flea market on Sundays.
The famous Brooklyn Ice Cream Factory is guaranteed to have long lines. Be patient or go up the street on Old Fulton Street and get some gelato. Your famous pizzerias will be along that street also.