If I host a Minecraft Server on a VPS will I have to leave my computer on 24/7 so the server doesn't close? I am hosting a Minecraft Server on a VPS just now and I open up the Minecraft Server and it starts generating the map and loading plugins etc. But if I close the tab for the Minecraft Server, the server will just shut down. I want to know how to host a Minecraft Server on a VPS but not leave my computer on 24/7.

You'll have to get a web host and pay for them to run your server. You can choose from all sorts of options.

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windows vps hosting - Free Setup - Managed VPS √ Dedicated hardware √ MS Hyper-V technology √

If you go with the cheapest OpenVZ VPS, which most people do, you'll have poor performance with Minecraft due to Java. You will need a Xen or KVM VPS, which is more expensive. You could also look into providers who specialize in game hosting, like All Gamer.

I would be skeptical of going with a hosting company that does not deal with online gaming but would feel more comfortable with one that specializes in online game server hosting.