  We develop a general theory of `quantum' diffeomorphism groups based on the
universal comeasuring quantum group $M(A)$ associated to an algebra $A$ and its
various quotients. Explicit formulae are introduced for this construction, as
well as dual quasitriangular and braided R-matrix versions. Among the examples,
we construct the $q$-diffeomorphisms of the quantum plane $yx=qxy$, and recover
the quantum matrices $M_q(2)$ as those respecting its braided group addition
law.
