  Using a global symbol calculus for pseudodifferential operators on tori, we
build a canonical trace on classical pseudodifferential operators on
noncommutative tori in terms of a canonical discrete sum on the underlying
toroidal symbols. We characterise the canonical trace on operators on the
noncommutative torus as well as its underlying canonical discrete sum on
symbols of fixed (resp.\ any) non--integer order. On the grounds of this
uniqueness result, we prove that in the commutative setup, this canonical trace
on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace
which is thereby identified with a discrete sum. A similar characterisation for
the noncommutative residue on noncommutative tori as the unique trace which
vanishes on trace--class operators generalises Fathizadeh and Wong's
characterisation in so far as it includes the case of operators of fixed
integer order. By means of the canonical trace, we derive defect formulae for
regularized traces. The conformal invariance of the $\zeta$--function at zero
of the Laplacian on the noncommutative torus is then a straightforward
consequence.
