Can I re-download an audiobook that I purchased from iTunes again? Apps purchased from the app store and songs purchased from iTunes can be downloaded again by going to the purchased section in the respective app and clicking the cloud button.

Neither app shows the audio-books I have purchased and when I look at a specific book I have already purchased in the iTunes store it shows the buy button.

Will I get charged again if I click the buy button or will it just re-download it?
Most of these older answers are incorrect now. In recent versions of iOS you can re-download all your audiobooks purchased through iTunes by starting the iBooks app, and going to the Purchased tab at the bottom right. From there you can choose what to redownload from My Purchases.