Best Solution (IMHO): Harold Prin's Joomla and MW authenication plugins (more below).

Many developers, designers, and open-source enthusiasts want to combine Joomla! -- the world's most popular CMS -- with MediaWiki -- one of the most powerful wiki engines (the software that powers Wikipedia).  Both systems excel at what they do: Joomla! at managing content and functioning as a base operating system for websites, and MW creating powerful and expansive documentation sites.  As good as these two systems are, it's amazing that relatively few good solutions tie the two pieces of software together. Most bridges to date either compromise security or include hacks that complicate updates of one or the other the systems.

I've tried the old Joomlawiki extension (even have a BETA produced for Joomla 1.5 not widely distributed), but it used a heavily modified old version of MediaWiki, far behind many of the best new features and compatibility with some of the better MW extensions.

More recently, I've worked with JTAG's MediaWiki Bridge.  It's a promising component that's not hard to install (as bridges go) with not much hacking.  But it doesn't work with PHP in safe mode, or with openbasedir set -- two common security methods I'd rather not bypass on my sites.  The lack of apparent support on their forum is also an issue.

The lack of adequate complete bridges means I'm using just authentication integration.  This allows the two platforms to use the same user base (allowing users one account for both systems), but does not combine them further.  There's a MW plugin called AuthJoomla that does this, but implementation is sketchy (last time I attempted to use this method, it was with Joomla! 1.0x, ate up two days, and never worked correctly).  Better, in my opinion is Harold Prin's Joomla and MW authenication plugins. They were easy to install and configure, provided one follows the instructions carefully (I missed the salt$ variable assignment in LocalSettings.php the first time).

It's rumored that jFusion is working on MediaWiki integration for their universal bridge extension for Joomla!.  Right now, it's pushed to the 2.0 version (or later), due out late this year (2009) or next.  Upon release, this method will allow seamless authentication and frameless inclusion of wiki content in your joomla site.  If jFusion's previous integration support is any indication (I've used their excellent support for Moodle, Magento, and Docuwiki, all with good results), this will become the method of choice in the future, especially if MW isn't the only non-Joomla! software you'd like to include in your site.

BONUS: Use DosIdea's MediaWiki Search Plugin for Joomla! to allow results from your MW install to show anytime someone queries using Joomla's inbuilt search function.  While installation is a little more involved that typical plugin installations, the results are worth it!