What is the sooty stuff on the stems of my viburnum?

A photo would be helpful, but black, sooty stuff usually means honeydew, and brown, rusty spots could be a number of things. Might be Viburnum scale, though that wouldn't cause the "rusty blotches". Are the rusty blotches sitting on top of the leaf like a deposit, or do you mean orangey brown dead areas of leaf?
Also, any evidence of anything underneath the leaf that shouldn't be there?

Sounds like 'sooty mold', which as Bamboo says, grows on honeydew left by aphids. Spray to get rid of them.