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Question
Do I have to be connected to the Internet when I play Steam games?
Answer
Steam has an "offline mode," in which any game that is fully updated can be played even when you're not connected to the Internet.
To use this mode, you must "exit" from Steam, do NOT use the "exit and logout" option. If you start Steam while not connected to the Internet, Steam will ask you if you want to use offline mode.
It's that simple. Note that in order to play a game in offline mode, its content must be fully downloaded first. You can check to see how much of a game's content has been downloaded by looking at its Properties page (right-click on the game's name in your "My Games" list).
Offline mode is particularly useful for LAN games and large-scale LAN parties, where Internet connectivity is not practical.