I've been told of two ways of running a downloaded .RUN file.

Note: Bear in mind that the following is all case-sensitive.

Through the terminal server

1. Download the .RUN file to your desktop
2. Open a terminal by going to Application > Accessories > Terminal
3. Type 'CD Desktop' to go to your desktop folder
4. Type 'LS' just to check the .RUN file is in this one
5. Type 'sudo sh nameoffile.run'

Through the Gnome GUI

1. Right-click on the file you downloaded
2. Select 'Properties'
3. Select the 'Permissions' tab
4. Tick the 'Execute' checkbox

Conclusion

The solution using the terminal worked for me but the second one using Gnome didn't.