Windows (Mac/Linux on the way): One of BitTorrent's biggest downsides is its reliance on central servers. Free program Triber is a truly decentralised BitTorrent client, meaning you can search for and download torrents straight from the peers themselves, without external servers.
The average downloading of a torrent requires accessing centralised servers, usually more than once. Not only do you need to find the torrent on a search engine like The Pirate Bay, but then you also need a tracker to manage the transfers and connect you to the peers that share the files. The problem with these servers is that sometimes trackers go down, rendering your transfers useless until they come back up — or, in worse cases, the trackers and search engines get shut down by the powers-that-be. Even trackerless DHT torrenting often requires the occasional server to get started.
The newest version of BitTorrent client Tribler completely changes the way you download torrents, doing away with these central servers. You can search for torrents from the in-application search box, which, instead of searching popular torrent sites, searches the connected peers directly. It still uses trackers to manage the downloads, but can continue downloading if and when they go offline.
Of course, it also has many of the advanced features you've come to expect from modern BitTorrent clients, like playing a video file in VLC while it's still downloading, support for magnet links, spam control and more. The only downside of Tribler at the moment is that it isn't quite as popular as other clients, so the number of available torrents is much smaller. You can probably still find more popular stuff on its network, but for now, you'll probably have to stick to the traditional methods for downloading more obscure files.
Tribler is a free download for Windows. The newest build, which contains all the above enhancements, is not yet available for Mac and Linux, but should be very soon (within the next week or so), so keep your eyes out. Also, if you're curious about how Tribler works and the other advantages of server-free torrenting, I highly recommend you read TorrentFreak's very informative article on the subject.