As previously announced, Twitter is now rolling out a new option which lets you download your entire history. Users are now allowed to download every tweet under the Twitter's API limit which is 3,200 tweets, however the option isn't yet available to all users.
If you're lucky, you can download history by going to your settings and request it. The only disadvantages is you have to wait until Twitter email's you with a zipped HTML file containing your history.
Twitter user Navjot Singh has briefly blogged about the process, explaining:
"Twitter informs that they will mail you the download link when the archive is ready. Yes, it's just like how Facebook's archive system works. Once you get the mail and download it you will get a zip file with archive in html form. Extracting it (sic) and you will see all your tweets sorted in calendar format."