Can fire happen in the absence of oxygen?

Subject says it all, but can you have a fire condition without oxygen present?

Well, according to the first definition, no.

Hydrogen will burn in fluorine, with a flame. So will many other materials.
Magnesium will burn in both nitrogen and carbon dioxide.

A fire is just a fairly fast and dramatic chemical reaction. Any fast exothermic reaction could produce similar phenomenon to what we call fire.