Why do monarchy still exists?

No particular reason in most cases other than tradition. I've got some British friends who love and adore the Royal Family even though they have no real power in the government.
Some people want to do away with it, but more people want to keep them. They do tend to bring in a boatload of tourist trade.
However I don't think that there would be any great race to replace any Royal family who suddenly becomes extinct. For example, if the entire British Royal Family all died tomorrow in some freak accident or epidemic, I doubt that there would be a massive manhunt for the next in line to the throne.