Is it easy to read information on your pages? Check your pages for color contrast.
Check color contrast online
Number of mobile users is growing each day, so make sure that your website is friendly enough for mobile devices.
W3C mobile checker
Google provides a number of rules to optimize your page load. And be sure they use it while ranking your website in search results.
Google Page Speed
There is also a number of parameters you may optimize with tool from Yahoo. Unlike Goole they have documentation that shows how score is calculated.
Use the service to find out what can you fix in server response headers to optimize page load.
Make sure you have the latest version of Joomla! If not, update it!
Download latest Joomla!
You may perform periodic checks to be sure that site is clean of malware.
When you develop or customize an extension be sure to protect your site from CSRF attacks.
Explanation and walkthrough
Rename htaccess.txt to .htaccess and make sure RewriteEngine is set to "On".
Check configuration.php file permissions. Set it to 644, or much better to 444.
If you still on Joomla 1.5.x, do not use the standard jos_ table prefix.
Do not promote that website is built on Joomla! and its version for security reasons.
Remove Joomla generator tag
If you go live with the site being established on development environment, a good idea would be to check /log and /tmp folders paths in "configuration.php"
You do not use them and do not update them. So they are potentially dangerous.
Vulnerable extensions list
Look in your website structure for staging/development files and delete them (.psd, .bak, IDE files, copies etc.)
Check most important SEO criteria and calculate your grade for every page of website.
SenSEO for Firefox
Make use of 404 page by offering useful links and site structure.
A more useful 404
Do you use other cool tools in Joomla! development and maintenance? Share your experience — we'll be glad to extend our checklist.