  We formulate a quantum generalization of the notion of the group of
Riemannian isometries for a compact Riemannian manifold, by introducing a
natural notion of smooth and isometric action by a compact quantum group on a
classical or noncommutative manifold described by spectral triples, and then
proving the existence of a universal object (called the quantum isometry group)
in the category of compact quantum groups acting smoothly and isometrically on
a given (possibly noncommutative) manifold satisfying certain regularity
assumptions. In fact, we identify the quantum isometry group with the universal
object in a bigger category, namely the category of `quantum families of smooth
isometries', defined along the line of Woronowicz and Soltan. We also construct
a spectral triple on the Hilbert space of forms on a noncommutative manifold
which is equivariant with respect to a natural unitary representation of the
quantum isometry group. We give explicit description of quantum isometry groups
of commutative and noncommutative tori, and in this context, obtain the quantum
double torus defined in \cite{hajac} as the universal quantum group of
holomorphic isometries of the noncommutative torus.
