The Windows Update site, and the newer Microsoft Update site, are designed to work only with Internet Explorer. The update sites depend on an ActiveX control which scans your computer to determine which updates are missing. Firefox does not run ActiveX controls.
This article describes other options for updating Windows.
Using automatic updates
Windows XP and Windows Vista include the ability to locate, download, and install Windows updates without having to browse to Microsoft's website. You can choose to have updates downloaded when they are available and either installed automatically, or held for your review. The Windows help system can guide you in choosing among the options. If you need additional assistance in setting up Automatic Updates, read these instructions on Microsoft's website.
Updating manually
Even with automatic updates enabled, you might want to visit Microsoft Update online from time to time for optional updates (for example, new versions of Windows Media Player). There are three ways to access the update site.
Using Internet Explorer
Internet Explorer 6 and 7 have an option to visit the update site using update.microsoft.com in the address bar and press Enter. (As with any download, it is a good practice to confirm that you are at the genuine Microsoft site before allowing any software to be installed.)
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Based on information from Windows Update (mozillaZine KB)