  The generalization of (super)integrable Euclidean classical Hamiltonian
systems to the two-dimensional sphere and the hyperbolic space by preserving
their (super)integrability properties is reviewed. The constant Gaussian
curvature of the underlying spaces is introduced as an explicit deformation
parameter, thus allowing the construction of new integrable Hamiltonians in a
unified geometric setting in which the Euclidean systems are obtained in the
vanishing curvature limit. In particular, the constant curvature analogue of
the generic anisotropic oscillator Hamiltonian is presented, and its
superintegrability for commensurate frequencies is shown. As a second example,
an integrable version of the H\'enon-Heiles system on the sphere and the
hyperbolic plane is introduced. Projective Beltrami coordinates are shown to be
helpful in this construction, and further applications of this approach are
sketched
