What are those dark spots on my ceiling?

I had a customer ask me today why she had dark spots on her living room ceiling. Specially in her corners (left pic). My first question was if she uses a wood furnace which in this case she did. Her furnace was stationed in the kitchen but the soot would spread throughout her vents. We got  lucky and it ended up being a easy fix. One coat of Killz™ Latex Primer (top right pic) and two coats of Sherwin-Williams™ Eminence ceiling paint. The result was a clean/white ceiling that looked like new (bottom right pic).