Backup all board files and the database.
Download the Full Package of the latest version of 3.2.x
Unzip to your desktop and open the phpBB3 folder.
Remove (delete) the config.php file, and the /images, /store and /files folders from the package (not your site).
Go to the ACP, Board settings, and make sure prosilver is set as the default style. If not, set it to prosilver.
Using FTP, delete the /vendor and /cache folders from the board's root folder on the host.
Via FTP or SSH upload the remaining files and folders (that is, the remaining CONTENTS of the phpBB3 folder) to the root folder of your board installation on the server, overwriting the existing files. (Note: take care not to delete any extensions in your /ext folder when uploading the new phpBB3 contents.)
If there are any install_config.php, *.lock or migration* files in the /store folder, delete them.
In your browser go to http://www.example.com/yourforum/install
Follow the steps to update the database and let that run to completion.
Via FTP or SSH delete the /install folder from the root of your board installation.
Done.
You now have a new up to date board containing all your users and posts.
Follow up tasks:
Update your language pack
Update your style
Only use a good FTP or SSH client, such as FileZilla, to transfer the files. (File transfer utilities provided in host control panels can cause problems.)