You've had Roundcube up and running for a while, and now there is a shiny new version out and you need to upgrade. Where to start?
Download an Updated Version
As with installation, you need to download the files for the newest version. The easiest way to get it is to visit http://www.roundcube.net and click on "Downloads" or just go to http://www.roundcube.net/download. Wait for it to download, and then unzip (using WinZip or "tar xvfz roundcubemail-0.1beta.tar.gz") in the current directory.
Now you should read UPGRADING and INSTALL files and check system requirements of the new version.
Backup, Backup, Backup!
I can't stress this enough: Back everything up!
First, make a copy of your Roundcube directory. This will help in case the new version does not work for you, you can always go back. If you are feeling daring, you can backup only your configuration files config/main.inc.php and config/db.inc.php (do not forget about plugins configuration).
Next, perform a database dump and save the file in a safe place. If the new version of Roundcube requires any changes to the database schema, you may need to restore this backup if you have to revert to the earlier version.
Check Your Configuration
Download your config/main.inc.php and config/db.inc.php files and place them in your config folder so you will not have to copy them over later.
Be sure to check config/main.inc.php.dist and config/db.inc.php.dist to make sure you have any new settings that may have appeared in the new version! Some people may prefer to copy the new .dist files and recreate their needed changes, but preferred way is to use installer (or update.sh script).
Upload a new copy
Use your favorite FTP/SFTP/SCP program to rename or remove the old folder and upload the new one.
After you're done you have one final step to do before you can start using your new version of Roundcube. Run bin/update.sh from the command line OR open http://url-to-roundcube/installer/ in a browser and choose "3 Test config". To enable the latter one, you have to temporary set 'enable_installer' to true in your local config/main.inc.php file.
If desired, you could also drop all of the tables and use the *.initial.sql file to fully recreate the database.
That should be all that is needed to upgrade. If you have problems, you can always restore the backups and try again, or ask for help on the forums. If you need help, be sure to take note of any errors that appear on your screen or in the log files.
There's also UPGRADING file in Roundcube package to which you should refer to. Wiki can be outdated.