XNA Game Studio Express 1.0 is not officially supported on Windows Vista and indeed, will give you some trouble if you just go ahead and try to install it. This guide will tell you how to install XNA Game Studio Express so it will run on Windows Vista with all features you get on Windows XP.
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1. Install Visual C# 2005 Express
Just run setup as you would under Windows XP. You can download the installer from this URL: http://msdn.microsoft.com/vstudio/express/visualcsharp/
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2. Install the Visual Studio 2005 Express Edition Service Pack 1
Now we're going to upgrade Visual Studio 2005 with its recently released first service pack. Download and install it from the following URL: http://www.microsoft.com/downloads/details.aspx?FamilyID=7B0B0339-613A-46E6-AB4D-080D4D4A8C4E
If you would run Visual C# 2005 Express now, you would be seeing the following notice which tells you to install yet another upgrade and run Visual C# 2005 Express with administrator privileges:

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3. Install Visual Studio 2005 Service Pack 1 Update for Windows Vista
You can download the update from this URL: http://www.microsoft.com/downloads/details.aspx?FamilyID=90e2942d-3ad1-4873-a2ee-4acc0aace5b6.
After installing this update, the notice displayed by Visual C# 2005 will change:

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4. Install XNA Game Studio Express 1.0
This step is a bit more complicated. Just running the installer will install XNA Game Studio Express, but leave us without the project templates (which are the only way to create new XNA applications). The trick is to run the entire installation with administrator privileges. Here's one way to do it:
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4.1. Open a new command prompt with administrator privileges

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4.2. Run the XNA Game Studio Express installer from this command prompt

If you don't know the path to the msi file, you can move it to your C:\ root directory and enter C:\xnagse_setup.msi instead!
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5. Run Visual C# 2005 Express with administative privileges
Even if everything seems to work without administrator privileges, remember Microsoft themselfes recommended to run Visual C# 2005 Express (and therefore XNA Game Studio Express) with administrator privileges. The most convenient way to do this is by setting "Run program as administrator" in the link properties:

There it is: XNA Game Studio Express 1.0, running on Windows Vista with all the project templates, the content pipeline and project extensions working!
