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Back up - Falkirk Bairn
You may well be able to restore the files to the new computer HDD but they will not necessarily work as the Operating systems could be different types (2000, XP or Vista) or even just different releases / versions of the OS.

The best plan is to copy all data (not programs) to the ext HDD and then copy the data back to the new computer - after you have installed the programs on the new system from original Software Discs you bought over the years.