Backlinks enable you to keep track of other pages on the web that link to your posts. For instance, suppose Alice writes a blog entry that Bob finds interesting. Bob then goes to his own blog and writes a post of his own about it, linking back to Alice's original post. Now Alice's post will automatically show that Bob has linked to it, and it will provide a short snippet of his text and a link to his post. What it all works out to is a way of expanding the comment feature such that related discussions on other sites can be included along with the regular comments on a post.
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