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Those who are VAC Banned cannot enter a VAC-enabled server. However, this only applies to the engine the game uses. For example, if you are banned on Counter-Strike or Half-Life, only games which use the GoldSource engine will be affected. You can still play games and connect to secure servers on games that do not use the same engine. In the past, those who were VAC Banned had all non-purchased items deleted from their backpack. In the current version of Team Fortress 2, users who are VAC banned do not lose any items in their backpack. However, all items are stuck inside of their backpack and cannot be traded away, nor placed on the Steam Community Market. These users are still able to purchase items off of the Mann Co. store, and use said items freely. VAC banned players are still able to play on all VAC-insecure servers on any game that the ban had an effect on.
VAC has several versions. For example, Team Fortress 2 uses version NN. VAC is module based. Most clients only see a fraction of all the modules the system has.
VAC bans are non-negotiable. The only way to get a VAC ban removed is if you contact Steam Support and they realize that you have been banned by mistake or something you did accidentally triggered Valve Anti-Cheat. You cannot simply lie, as Steam has all the info about how and why you were banned. If your alibi matches up with how you were banned, and it appears that the ban was wrongful/accidental, your ban will possibly be removed. There is no guarantee.
As with any Anti-Cheat, VAC is not perfect. Certain hacks may be able to slip through the cracks, if only for a while. Once VAC has detected a user hacking, it may take from a few days up to a few months before said user is banned from secure servers. This is done to prevent the cheat developer that made the hack from pinpointing the exact cause of the ban. Additionally, VAC runs differently on Linux machines, making it much less effective and easily bypassable. Valve has not talked about any fix for this as of yet, but in a recent Q&A with Valve CEO Gabe Newell, he has said that Valve Anti-Cheat is one of the things that they will be working on improving in the future.