Twitter has started rolling out the option for users to download their Twitter archive. That file will contain all your tweets up until the time you download it, arranged chronologically and searchable. This "data liberation" feature is already available from Google and Facebook, so while you're at it, you can download your data for those services as well.
The "request your archive" option isn't there for everyone yet, but a few Twitter users have seen it already at the bottom of their Settings pages. Twitter user @nspeaks says that after you request your archive, Twitter will email you the download link to the zip file when it's ready (much like Facebook's system). In the zip file is a Twitter index.html file with your archived tweets, as well as a chart of your Twitter activity by year.
What's the point? Well, it's not just about navel gazing, but about having ownership of your data and the ability to port it elsewhere if you so desire. Also, because Twitter only lets you search so far back (maybe a few days or week if you're lucky), having the Twitter archive is useful for finding those links and other information you've shared.
As noted above, Google and Facebook both offer the ability to download some of your data. You can "liberate" your data from Google with its takeout products for Google+, Google Docs, Picasa Web Albums, and more. See The Data Liberation Front for more information and instructions.
For Facebook, you can request a copy of your data from your Account Settings > "Download a copy of your Facebook data" > Start My Archive. The archive includes your timeline information, such as posts and photos you've shared, as well as chat history, check-ins, and more .
Hooray for data liberation!