In December 2012, Twitter announced that they were opening up their archive and making your past tweets available for download. You can access this by visiting twitter.com > click the cog in the top right hand corner of the screen > select "settings" > scroll down to the bottom of the page > select "Request your archive".
Once your archive is downloaded upzip the file (right click on the file and select "Extract All") and open. Double click on the browser option and your archive will open up in a web page. You can view your tweets month by month, search for specific words or hashtags and usernames. You can also interact with your old tweets just as you can now.
It has been noted by Claire (@BumpWearClaire) that you can only view the archive of your existing @username. If you have changed your name in the past you currently cannot access/download that archive. I'll update this post in the future if that facility becomes available.
I have also spotted some users mentioning or retweeting old tweets. Does this mean we are in danger of creating a Twitter
And I shall end on the type of information that hugely excites the Geek in me... I was the 17,636,207th user to sign up to Twitter.