Twitter announced they are rolling out a way to download all your historical tweets to your computer. A move that gives them good rep in the community and makes them look better than Facebook and even better than Google (although Google does let you download a lot of your data).
The feature, not yet active for me, is under your Twitter settings. You scroll to the bottom and you should see a button to archive your tweets. Here is a picture:
Once they are downloaded, it will store the files in a folder, which you can then open in your browser or editor of your choice.
Here is a picture of how you can scan your archived tweets by date or even search them:
Danny Sullivan has a nice step-by-step walk through on how to do all of this at Marketing Land.
I await a slip up and way to download and archive tweets you do not own.
Forum discussion at WebmasterWorld.