Pick pocketing and bag snatching do occur, but they are very rare and basic precautions such as not leaving your wallet or purse open on the table showing a giant wad of cash should be observed.
There have been incidents of physical and verbal sexual harassment of women in the UAE, as well as sexual assault. In particular, underground walkways near the Abu Dhabi corniche has had multiple incidents of sexual assault on women walking alone, so be extra careful in this area and do not travel alone if you can avoid it.
Taxi drivers have been reported as a source of harassment and it is advisable for women to sit in the back seat, and not to engage in the general sort of conversation one may engage in with taxi drivers in western countries. For some reason, even polite conversation such as "Yes, I am very well thank you and how is your day?" can be seen as propositioning the taxi driver. We have absolutely no idea how that mentality arises, but nonetheless it is there.
If you need to catch a taxi, get the hotel or restaurant to make the call for you especially if you are a solo female traveler.
Cultural differences in the UAE and western nations sadly mean that women may need to dress a little differently than they'd like and all female travellers should be keenly aware that what is general politeness in a western country may be taken as an invitation for unwanted attention in the UAE.
What may be acceptable on the beach isn't acceptable everywhere else in the UAE. Dress modestly by covering your shoulders and wearing a skirt or pants below the knee when traveling around, the locals will appreciate the respect for their culture. Plus it protects you from the hot sun and potentially thawts any sleazy types. Bear in mind, that outside of the UAE things can be much more conservative so it pays to keep the peace!
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If it has somehow escaped your attention, the UAE legal system is largely determined by Sharia law, and Sharia law prohibits alcohol. So, can you get a drink while visiting UAE?
Use and possession of drugs are serious crimes in the UAE with very little leniency shown. Even prescription and over-the-counter medications could land you in trouble.