How Do You Go About Mending Your Reputation?

A lot of things in life depend on your reputation. Employment, relationships and many other things could be easier or harder to obtain if you have a poor reputation. Many people spend large amounts of time doing activities that can build or at least not hurt reputations that people can respect. However, what happens if all of a sudden, your reputation is under threat? Consider the following 2 cases;

1. You are accused of doing something you genuinely did not do and yet people are convinced you did it. How do you recover from such a hit?
2. You actually did do something shameful. How do you recover from that?