capital city

A city that is a seat of government.

A city (or cities) much larger or more important to a country than all others, regardless of the actual seat of government (cf., Moscow and Saint Petersburg regardless of moves of the Russian government; Amsterdam despite the Dutch government having been situated in The Hague since 1588).

(Australia) The major metropolitan areas of Australia (Sydney, Melbourne, Brisbane, Perth, Adelaide), all of which happen to be state capitals, and by extension the major metropolitan areas of other countries; note that other state capitals Hobart, Darwin, and national capital Canberra are often excluded.