Check if a process is running in 64 or 32-bit mode?

Can we assume this is a Windows question? XP x64, Vista, 7?

In Windows 7 I believe a 32-bit application running in a 64-bit OS has a *32 after its name in the Task Manager...

Just to add to this old answer, yes use Process Explorer. You can right click on the columns and select Image Type to show which processes are 32 bit and which are 64 bit. Run Process as Administrator to see some services.

http://live.sysinternals.com/procexp.exe

NO NEED EXTERNAL PROGRAMS!

Open Task Manager > Details, right click on columns bar Select Columns and check Platform:

(Windows 10 and 8)