Can I upgrade to the new Ubuntu 10 from Ubuntu 9.10 without formatting my computer?

If you don't have it already, you gonna need a free partition on your harddrive to install it beside your current OS. If you need to create a partition, Windows Seven (if you have Windows 7) allow you to shrink a partition so you'll be able to create some place for Ubuntu.

If you already have it , just go in Updater Manager panel and you'll have an option to upgrade to 10.04 LTS on top of it.

More details here : https://help.ubuntu.com/community/LucidUpgrades#Upgrade%20from%209.10%20to%2010.04%20LTS

Update : (thanks to th3dude )

Could check this out too that outline exactly what I'm talking about :

Update to Ubuntu 9.10 from Ubuntu 9.04

You can use Update Manager as specified in Julien's answer, or from a terminal you can use:

sudo do-release-upgrade

See http://www.ubuntu.com/getubuntu/upgrading

Yes, do ALT + F2, type "update-manager -c", and the update manager will offer you to upgrade your Ubuntu version.