Mobile (cell) phones: GSM (900/1800) and 3G (CDMA, UMTS) is used in Ukraine. This system is compatible with mobile phone networks used everywhere apart from parts of the Americas and US dominated parts of Asia.
If you have an unlocked GSM phone, you can get an Kyivstar, MTS or life:) (Astelit) SIM card for a few dollars at street vendors which will give you a local number and free incoming calls. Note that most of those cards don't have money on their account so you may want to buy a payment card when you buy a sim card. If you don't have an unlocked phone already, new ones can be had for USD30-40 and a touch cheaper if you buy a pay-as-you-go sim card at the same time. Incoming calls are free in Ukraine so in extremis you can just SMS/text a request for a return call for a small charge.
If you want to use 3G connection, you can get OGO! (ex-Utel) for UMTS and PeopleNet, CDMAUA or Intertelecom for CDMA, for mid 2011 last three operators don't have English version of site.
If you are roaming in Kiev, SMS messages do work well. They are confirmed to work for most foreign networks. Do note that the size of the country and the relative low population densities of rural areas means that sometimes there might be 'black-spots' where mobiles will not work. But of course these are away from the main cities/urban areas (and most of the main arterial road and rail routes also have reasonably consistent call signals).
If you are trying to call the US from your GSM phone, you may find that the access numbers for your calling card are blocked. Plan ahead and sign up with a callback service (such as UWT **warning, lead-time required**) before you start your travels and you can provoke them to call you (at much better rates) when you need to make a call.
The easiest way to maintain internet connectivity if you use your own laptop is to buy a 7-day unlimited Lucky Internet callback card. They are about UAH36 at the street kiosks. When you dial in, you will be initially firewalled off from everything until you activate by visiting their website
You may also buy wireless internet access for your laptop for about 10 UAH per day. for details or just google "wireless internet in Ukraine".
Internet cafes have a good service. They usually have different types of computers with varying prices.
Near the metro station on ul Khmelnytskoho (on the left side at a corner) there is one that is very good, open 24 hours non stop. The cheapest computers cover your basic needs, the most expensive ones are usually for hardcore gamers.
Also most foreigner-friendly cafés (see "Drink" section above) and a lot of fast food restaurants (including McDonald's) offer free Wi-Fi. Some require password to use their access point, ask waiter to get it.
Tr'okhsvyatytel's'ka St, 6
69, Frunze str.
24 Volodymyrska Str
Lavrskaya St., 9
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