Ubuntu 13.10 ships with Gnome Desktop 3.8.x. Thankfully the GNOME team make it easy to install/upgrade to GNOME 3.10 on Ubuntu 13.10 Saucy. The GNOME3-Next PPA will contain GNOME 3.10 packages that are considered stable.

NOTE: At the moment of writting this tutorial. The PPA only contains few packages and it will continue adding gnome 3.10 packages in coming weeks.

GNome 3.10 lock screen

To get started, press Ctrl+Alt+T key combination on your keyboard to open terminal. When it opens, run below command to add the GNome 3 Next PPA:

sudo add-apt-repository ppa:gnome3-team/gnome3-next

Then update package lists:

sudo apt-get update

Install GNome 3.10:

sudo apt-get install gnome-shell ubuntu-gnome-desktop