All of a sudden I realised that I hadn't had any Windows 7 updates in a while - so I decided to check "Windows Update" (from the Start Menu - just type 'update' in the search box) - and discovered that there was a problem.
I clicked "Check for Updates", and the page changed to the "Checking for Updates" page, with the moving progress bar. And there it stayed, for hours, doing nothing. No error, no timeout, no nothing.
So I dug a little deeper - the System Tools Event Viewer showed a System Event error:
"The Windows Modules Installer service terminated with the following error: The specified module could not be found." - Event ID 7023
What's more, if I tried to start the "Windows Modules Installer" service from services.msc, I got a similar error.
Hmm - something was wrong with "Windows Modules Installer" - whatever that was. Anyway - the long & short of it is, I eventually found a page with the solution. It's here: http://www.vistax64.com/tutorials/282940-windows-modules-installer-error-126-fix.html
Yes, it's a Vista page, but the "Fixing Key" part was what sorted it for me.
The error I had was that the version number in C:\Windows\Servicing\Version was different to that in the Registry key in
HKLM\Software\Microsoft\Windows\CurrentVersion\Component Based Servicing\Version
The web-page I've linked to shows you how to change the registry to have the right version number. Once I'd done that, BANG! Windows Update swung into action without me having to anything else.
Phew. I was just about to line up a complete re-install of windows...