The terms "Old World" (Africa, Asia, Europe) and "New World" (the Americas) originate from a Eurocentric, post-1492 worldview.
Australia belongs to neither. Geographically and geologically, it is part of the continent of Oceania** (or Australasia), which includes New Guinea, New Zealand, and the Pacific islands.
In biogeography, the Wallace Line separates Australasian fauna from Asian (Old World) fauna, underscoring its distinctiveness.