Hi 1stsolo65,

Yes, you can uninstall Norton 360 from your computer and then install it on your new computer without losing your licence or subscription.

Please do the following for a smooth transfer:

Depending on which edition of Norton 360 you have got, please download the correct edition of Norton 360 version 4 from the appropriate link below and save it to your desktop:

Norton 360 Standard Edition version 4 (English)
http://buy-download.norton.com/downloads/CLT/N360/US/2010/4.1/ESD/N360-ESD-17-6-0-32-EN.exe

Norton 360 Premier Edition version 4 (English)
http://buy-download.norton.com/downloads/CLT/N360P/US/2010/4.1/ESD/N360-PREMIER-ESD-17-6-0-32-EN.exe

You will also need your Norton product key in order to activate Norton 360; during activation, your subscription days will be picked up automatically by Norton 360. You can fetch your product key from your Norton account:
http://www.mynortonaccount.com/
You can also find your product key on the CD sleeve.

Install Norton 360 on your new computer and run LiveUpdate immediately after activation in order to obtain all the updates. You will receive an engine update as well and you will be asked to reboot your computer.

During activation, a notification may pop up saying that you have exceeded the number of allowed activations. If this happens, please contact Symantec Customer Support, and they will reset your activations for you. You can chat with them for free over here:
http://www.symantec.com/norton/support/contact/contact.jsp?pvid=cs

Important note! Please remove any preinstalled antivirus software from your new computer before installing Norton 360. Most antivirus software vendors provide a free removal tool on their respective websites to ensure a clean uninstallation of the antivirus product.