If you've tried upgrading to CiviCRM 3.1.3 in Joomla, you may have run into the following error:

DB Error: no such table
Database Error Code: Table 'your_database.civicrm_contact_type' doesn't exist, 1146

The solution is really simple, just follow these instructions, expertly provided by Deepak Srivastava over at the CiviCRM community forums:

Workaround - 1
As soon as you see the error, doing a page reload should bring the upgrade screen back. And hitting upgrade button should work normally.

- OR -

Workaround - 2
Before installing the new codebase increase the session lifteime, so that session doesn't expire between installing the new codebase and hitting the upgrade.
Note: Session lifetime could be increased from Global Configuration >> Session Settings >> Session Lifetime. Change the session timeout back to previous one, once you done with upgrade.

Other than that, follow the official instructions, and you should be home-free!