The usual "don't be stupid" advice seems to be adequate. Kiev is a generally open and friendly city and stays lively until at least 11PM in most districts.
Avoid drinking the water from the tap — bottled water is cheap and available everywhere (Morshinska/Моршинська, Mirgorodska/Міргородська is good).
If you are female, and especially if you are traveling alone, try to take a taxi instead of public transit after 9PM These are prime drinking hours and the metro and marshrutky may be crowded with drunken men. This is particularly true on the weekends. Ask a local English-speaker to call the taxi for you and get the amount of the fare in advance; drivers may greatly inflate the fare once hearing your accent.
Robberies and scams on tourists are fairly common in Kiev. The best approach is to be extremely selfish and ignore anyone who approaches you. Avoid eye contact with suspicious looking people. If you do get caught up in a scam (such as the infamous wallet scam or the "Look, I've just found money" scam or even if you are stopped by someone claiming to be a policeman), simply ignore the person and walk away, indicate that you want to call your embassy or go to the next police station to get the problem sorted. That will usually shake the person off.
If you are leaving your baggage in the station, it is better to leave it with the guys in person rather than use a locker. Stories have been heard of people 'assisting' with the locker and overseeing the code, then walking off with the bag afterwards.
On the metro, always keep your belongings securely zipped as close to your skin as possible. Pickpockets are highly organised and often in gangs that know what they are doing.
There are occasional (rare) reports of visitors being shaken down by corrupt officials, often customs officials. Naturally, the best protection is to make sure that you stay on the correct side of the law and, if there is any question, to keep your cool and not become argumentative. It seems that the cost of an error is surrendering the object in question and paying a "fine." The officials are skilled at ensuring that people who argue miss their flights. Making, or giving the impression of making a phone call to your country's embassy has been known to clear up "problems" quicker than actually paying the "fine" --- or pretend to have a very late flight.
Walking around in the tourist areas (e.g. Maydan) at night you will no doubt find yourself stopped by groups of police who will stop you and go through an overly theatrical display of asking to see your passport and other documents, and sometimes even search you for drugs or other prohibited items. To deal with this, firstly always carry your passport, not doing so will give them an easy excuse to extract a bribe. Secondly, making it clear that you speak absolutely no Russian may help. Lastly, don't given them any excuses whatsoever to threaten you with arrest - just because the locals drink in the street doesn't mean you should. Don't cross the road except at authorised points. Try and follow the rules.
Some thieves like to abuse new tourists, for example, by playing plainclothes cop. They are rarely aggressive. They will go to you only if you're walking alone and don't look too familiar with the town. A bit of resisting usually shakes them off (but not too much since you never know).
There is still some corruption in Ukraine; some services might openly ask you to bribe them to process your request, and denying it might make them refuse to help you.
The people are very tolerant and it is only reasonable to assume that they expect the same in return.
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