Manhattan and New York generally have experienced a major falloff in crime during the past decade - in fact, for the past few years, New York City has been the safest major city in the United States - so there is no need to be afraid to walk most of the streets day and night and take the subways and buses. However, precautions should still be taken.
Keep your wits about yourself. Try your best to know or at least look like you know where you're going, particularly in areas which are deserted or otherwise feel potentially dangerous to you. Keep your wits about yourself by being aware of what's happening around you on the street, where the open shops are, where you may have spotted any police officers around, etc. Do not hesitate to calmly increase your pace, alter your route, or cross to the other side of the street if you sense it might be the safest course of action.
Beware of pickpockets. During the holiday season, pickpockets like to target shoppers near tourist attractions such as Times Square, 42nd Street, and Macy's, and anywhere where there is a crush of crowds. In order to foil pickpockets, never put your wallet or anything of value in your back pockets, but only in your front pockets. If you use a purse, make sure it is tightly closed and hold on to it. And when you sit down, such as in a restaurant, be careful to keep your valuables in places where an opportunistic thief would be hard pressed to snatch them and run.
Traffic hazards. Manhattan is in certain ways a pedestrian's paradise, but beware that traffic regulations are not always obeyed to the letter. Watch for aggressively turning cars and bicyclists riding the wrong way on one-way streets or on sidewalks. The problem is not constant, but these things happen often enough for them to be worth keeping in the back of your mind while walking on the streets and sidewalks. Also, you'll note that jaywalking is commonplace among New Yorkers, but it can be hazardous to those not experienced in judging the speed of oncoming cars. So do not blindly follow a local, for there's a chance you'll be staring at the headlights of a car if you are not careful.
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Throughout Manhattan, open WiFi access points are abundant, including many parks and squares such as Bryant Park and Union Square. Starbucks now offer free internet, and some stores such as Apple SoHo and Tekserv offer free wireless Internet to customers.
All of the many branches of The New York Public Library (http://www.nypl.org) offer free internet access to anyone with a photo id or NYPL library card.