Summary: Determining whether or not your process or operating system is 32 or 64 bit is very easy with the aid of a tiny tool, and knowing where to look.
How do you tell if an operating system is a 64-bit or 32-bit?
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I'm going to expand that to include both hardware and software.
In other words, how to tell if you have a 32 bit or 64 bit capable system, and then whether or not you have a 32 bit or 64 bit operating system running.
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You could, of course, download and run the Windows 7 upgrade advisor, I suppose, and it'll tell you a lot about your system, including whether or not it'll support the 64 bit version of Windows 7.
But that's overkill.
Download, instead, Securable, a tiny little utility from grc.com. Run it, and you'll get the answer about your hardware. Here's mine:

As you can see, my CPU can handle 64bits even though I'm currently running a 32 bit version of Windows XP.
To find out what your operating system supports, right click on My Computer, in Windows XP and click on Properties:

If it doesn't say "64 bits", then it's 32.
In Windows 7 right click on "Computer" in your start menu, then click on Properties:

The resulting screen shows exactly which version you're running.
Related:
Should I consider the 64 bit version of Windows 7? With Windows 7 imminent and 64 bit capable hardware more common, many people are wondering if it's time to make the switch to 64 bits with Windows 7.
Can you have too much RAM? Is 64bit just a marketing ploy? 64 bit machines and 64 bit support are becoming more common, which begs the question: is it hype, or is it worth it? And how much RAM is enough?
Why doesn't Windows show all 4gigabytes of memory I have installed? Installing the maximum 4 gigabytes of RAM into your computer may not result in all 4 gigabytes being available. We'll look at why.
Article C3916 - October 30, 2009
Regarding how to know whether it is 32-bit or 64 bit, you mean to say that when we right click on My Computer, in Windows XP and click on Properties and if it doesnt show the readings we can 100% confirm that it is running on 32-bit . Correct me If I'm wrong
Posted by: RoshanAshraf at November 3, 2009 6:08 AMSlight correction for opening system properties: 1... open start menu...2...Right click "my computer in the start menu! (works in XP too!)
Posted by: Joe F at November 3, 2009 6:53 AMin Vista Home Premium I do this:
Posted by: Ken Laninga at November 3, 2009 8:23 AMStart
Computer
System properties
I ran "Securable"; it did say my processor was capable of running 64 bit, but I have heard that my mother board has to be compatible with 64 bit also. How can I determine that? I currently have Vista Home Premium 32 bit and would like to switch to Windows 7 64 bit.
Posted by: Kitty Ashenbrenner at November 3, 2009 1:56 PM