Although it's extremely hard to ban a book from publication outright, it's fairly easy, and fairly common, to ban books from public schools and public libraries in the United States. This is usually a decision for local government, or even bodies such as PTAs, so you may find a book banned in one town, but not in the next one over. The thing that might surprise you, or not surprise you depending on your views of the US, is that some very tame books have been banned over the years for bizarre reasons. Check out some of these books that have been banned at one point or another.
'Where the Wild Things Are' is a childhood classic for many, but it has been banned for being too dark.
'The Bridge to Terabithia' has been banned because some claim themes of death and isolation are unsuitable for children.
'Lord of the Rings' has been banned for reasons including its frequent and positive portrayal of smoking tobacco.
Joseph Heller's classic 'Catch-22' has often been banned for explicit themes and bad language.
American classic 'The Great Gatsby' has been banned due to F. Scott Fitzgerald's characters being too promiscuous, and drinking and partying too much. 
'Gone With the Wind's' positive portrayal of slavery and pro-confederacy stance has earned it a banning in many parts of the US.
You might think that Shel Silverstein's 'The Giving Tree' is harmless, but it has been banned for portraying a selfish and manipulative child as its main character.
Mary Shelley's 'Frankenstein' may be a classic, but over the years it has been banned on many occasions. Perhaps for giving man the power to give life.
In some areas, The 'Da Vinci Code' has been banned for its anti-catholic views.
You might not think that Chaucer's 'The Canterbury Tales' could elicit much trouble. However, it is actually quite raunchy and sexually explicit (albeit in Old English, so most people don't understand the dirty jokes), which earned it a banning in the US up until as late as 1995.
'Brave New World' has been banned for depicting drug use and promiscuous sex.
'The Diary of Anne Frank' has sometimes been banned due to violent content and situations, and graphic passages. In the US, some have even tried to ban it on the grounds of being pornographic, but we don't think they have a leg to stand on there.
'Animal Farm' by George Orwell has been banned because of its political views, as well as its portrayal of highly-intelligent talking animals. 
'Alice's Adventures in Wonderland' has been banned for having animals that can speak and reason as well as humans. And, perhaps also the suggestion of drug use.
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