Deep cloning objects

Question

I want to do something like:

MyObject myObj = GetMyObj(); // Create and fill a new object
MyObject newObj = myObj.Clone();

And then make changes to the new object that are not reflected in the original object.

I don't often need this functionality, so when it's been necessary, I've resorted to creating a new object and then copying each property individually, but it always leaves me with the feeling that there is a better or more elegant way of handling the situation.

How can I clone or deep copy an object so that the cloned object can be modified without any changes being reflected in the original object?