How to install XNA game studio on Visual Studio 2012?

Is it possible to create XNA games using Visual Studio 2012?

Yes, it's possible with a bit of tweak. Unfortunately, you still have to have VS 2010 installed.

1. First, install XNA Game Studio 4.0. The easiest way is to install the Windows Phone SDK 7.1 which contains everything required.

2. Copy the XNA Game Extension from VS 10 to VS 11 by opening a command prompt 'as administrator' and executing the following (may vary if not x64 computer with defaults paths) :

   xcopy /e "C:\Program Files (x86)\Microsoft Visual Studio 10.0\Common7\IDE\Extensions\Microsoft\XNA Game Studio 4.0" "C:\Program Files (x86)\Microsoft Visual Studio 11.0\Common7\IDE\Extensions\Microsoft\XNA Game Studio 4.0"

3. Run notepad as administrator then open extension.vsixmanifest in the destination directory just created.

4. Upgrade the Supported product version to match the new version (or duplicate the whole VisualStudio element and change the Version attribute, as @brainslugs83 said in comments):

   <SupportedProducts>
     <VisualStudio Version="11.0">
       <Edition>VSTS</Edition>
       <Edition>VSTD</Edition>
       <Edition>Pro</Edition>
       <Edition>VCSExpress</Edition>
       <Edition>VPDExpress</Edition>
     </VisualStudio>
   </SupportedProducts>
   

5. Don't forget to clear/delete your cache in %localappdata%\Microsoft\VisualStudio\12.0\Extensions.

6. You may have to run the command to tells Visual Studio that new extensions are available. If you see an 'access denied' message, try launching the console as an administrator.

   "C:\Program Files (x86)\Microsoft Visual Studio 11.0\Common7\IDE\devenv.exe" /setup
   

This has been tested for Windows Games, but not WP7 or Xbox games.

[Edit] According Jowsty, this works also for XBox 360 Games.

[Edit for Visual Studio 2013 & Windows 8.1] See here for documentation on installing Windows Phone SDK 7.1 on Windows 8.1. Use VS version number 12.0 in place of 11.0 for all of these steps, and they will still work correctly.

On codeplex was released new XNA Extension for Visual Studio 2012/2013. You can download it from: https://msxna.codeplex.com/releases