Lord Ganesha has a broken tusk. As per legend, when Lord Ganesha was writing the Mahabharata, the feather he was writing with broke. So, to stick to the condition of writing continuously, Ganesha broke his tusk and wrote with it!

Another popular legend is that Lord Parashurama cut off one of Ganesha's tusks with his divine weapon, Parsu. Because Ganesha did not allow Lord Parsuram to enter the Shiva abode and meet Lord Shiva!