Rust is just corrosion when it happens to iron (and iron based alloys, steel and such). On copper it's corrosion is called 'petina' or 'verdigry', on silver it's called 'tarnish'.

It can also be called oxidasation.

But it's all the same thing, metals gaining oxygen atoms thus changing their composition. Mostly for the worse, but sometimes for the better, Aluminium for example when it oxidises forms a layer of oxide that's harder than the aluminium and prevents further corrosion.

same with chrome and stainless steel,

Ever seen what happens to aluminum if you put mercury on it? it corrodes at an astounding rate as the mercury destroys the oxide layer protecting it so it corrodes very fast, hence why you cant take mercury thermometers on aeroplanes (true story)