keithold covered the minimum specs - here's some experience for you.

My 3 year old laptop runs it just fine, though it's specced not unlike a relatively new system (1.83Ghz Core Duo, 2GB ram, only 80GB hard disk, x1400 graphics) - it runs 7 very well.

I have a friend that runs 7 on his MSI Wind.  That's a single core 1.6Ghz Atom, 1GB RAM, and a tiny 1.8" 60GB hard disk.  I've used it, and while it isn't quite as snappy as XP, it runs it just fine.  I will confirm through experience that Vista would be sluggish on it - all of the Atom/1GB systems I've touched were inadequate for Vista.

Here's a guess as to what a decent, realistic system for Windows 7 might be:  Pentium 4 at >2Ghz or AthlonXP 2100+ or better, 1GB RAM, a basic DX9 compatible video card to use Aero (If you don't have one, you can get one that'll be plenty for $30-40 or so, I can suggest something for you if you'd like).  If the computer is <5 years old and is somewhat reasonably equipped, you're seriously just fine.  Anything modern at all is fine.

I'd still recommend upgrading to 2GB of RAM if possible, which isn't exactly expensive these days.
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You can run it on pretty much any computer sold within the last 4 or 5 years, with the only upgrade needed being the RAM to reach the 1GB minimum.
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Windows 7 can run on almost anything, it's just a matter of how well.

The worst computer you could get away with would have 30GB of HDD space, 512MB of RAM, and a CPU along the lines of a 1+GHz P3... it'd be slow but it would work without problems, and with painful load times.

To run smoothly you'd need 1GB of RAM, a non-painfully slow CPU such as the 1.6GHz Atom, and a GPU capable of DirectX 9* so Aero would work... a fast hard drive would be nice so load times are minimal since there wouldn't be a lot of ram for caching programs(hyperfetch).

7 is less demanding than Vista so anyone with one of those bottom-end computers with Vista preinstalled would benefit switching.

*You also need 128MB of VRAM... not like any GPUs have that little stuck on nowadays unless they're integrated, in which case you just have to set it.
Source(s):
Firsthand experience... I put 7 on the worst possible computer mentioned and it did function... the second system mentioned is me trying 7 on a $200 netbook with an old 100GB 77200RPM Hitachi HDD installed.

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