Thanks 'spaced' but that doesn't work, tried all incl 'snap to grid' - seems this is a problem a lot of people get. Type 'desktop icons keep moving' in Google and see how many are getting this trouble. Some people are getting all sorts of answers but none seem to work. It is an irritation but nevertheless one that would save quite a bit of time if resolved. Thanks again.
If you are running Windows XP, Right Click on your desktop, select "Arrange Icons By" and you should see "Auto Arrange" on the menu. If this is ticked, your icons won't be under your control. Untick this and they will stay where you leave them.