  We survey some results on homological dimensions of the algebraic, complex
analytic, and smooth quantum tori. Our main theorem states, in particular, that
the smooth and the complex analytic quantum n-tori have global dimension n.
This contrasts with the result of McConnell and Pettit (1988) who proved that,
in the generic case, the algebraic quantum n-torus has global dimension 1. In
this connection we also formulate some general theorems on homological
dimensions of nuclear Fr\'echet algebras.
