Righty. Thank you all so much for your invaluable help and advice.
Norton's now run out, so have uninstalled it using the Norton tool, and then did a search on my lappy for anything Norton, and deleted a couple of little notepads and word docs.
I then have literally just now installed Avast - all very smoothly and nicely - hopefully it'll do the job.
My next question is with my windows firewall switched on, do I need Zonealarm as well?
Or will I be okay with just Avast and Windows Firewall?