If you are a tourist and come to the country for 1-2 weeks, then you may need mobile Internet for your phone, tablet or laptop. Good news: it's possible to buy mobile Internet without any documents, i.e. without contract subscription. All of the following providers have prepaid tariffs.
Be sure to connect and check the Internet at the point of sale, just because the process can be tricky for foreigner.
To buy any of these sim cards, look for logos on the streets or find the list of shops on the websites.
Data updated in September 2016.
Coverage - maximum
Tariffs
Market leader. If you are a tourist, then buy either Vodafone or Kyivstar.

Coverage
Very simple tariffs
One of the two market leaders, we are using the Internet for this operator.

Coverage (only in Ukrainian and Russian)
Tariffs (only in Ukrainian and Russian)

Coverage (only in Ukrainian and Russian)
Tariffs (only in Ukrainian and Russian)

Coverage
Tariffs (only in Ukrainian and Russian)
Home internet is well developed in Ukraine. If you've been living in the country for a long time, everything is simple: just find out which providers offer Internet in your house (ask your neighbors or see ads near the entrance and on the street near the house). As a rule, in big cities there are several providers in every house. Their tariff plans are almost the same. You can get an unlimited channel from 2 Mbit for about $12-15 per month. For $20-25 per month you can get quite a good channel: 20 Mbit and more.
Unlike Europe, the speed of home Internet in Ukraine can be quite high. 100 Mbit and more is not fantastic. I believe that the main reason is the low number of subscribers at the moment in the presence of good channels. In Germany, for example, full 10 Mbit is rare (70% of the population uses the Internet), while in Ukraine it's a pretty weak Internet (20% of the population uses the internet). Moreover, in Ukraine the traffic is divided into external and internal (UA-IX), which is not typical for European countries. All websites located in UA-IX have excellent channels and high speed. In a big city normal download speed is about 2-3 MB/sec even in the evening.
You can be connected to the Internet in several ways, usually via a DSL modem or a network cable that is inserted directly into the computer. I cannot say which option is better, we have both types of connection in our apartment and see no difference.
The connection process is fairly simple. You call the provider, name your address and agree the time of a serviceman's arrival. You need to sign a contract and show your passport. Monthly payments are usually arranged via the provider's website.
Largest providers of Ukraine:
http://volia.com/rus/internet/packages/ - only in Russian and Ukrainian
http://en.ukrtelecom.ua/services/private_clients/internet/about/rates - in English
http://www.kyivstar.ua/ru/home-internet/tariffs/ - only in Russian and Ukrainian
Sites with a list of providers of Ukraine (only in Russian)
http://www.providers.in.ua/catalog
http://ipkiev.pp.net.ua/dir/
http://www.broadband.org.ua/