A universal algorithm to construct N-particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir element is presented. In particular, this construction shows that quantum deformations can be interpreted as generating structures for integrable deformations of Hamiltonian systems with coalgebra symmetry. In order to illustrate this general method, the $so(2,1)$ algebra and the oscillator algebra $h_4$ are used to derive new classical integrable systems including a generalization of Gaudin-Calogero systems and oscillator chains. Quantum deformations are then used to obtain some explicit integrable deformations of the previous long-range interacting systems and a (non-coboundary) deformation of the $(1+1)$ Poincar\'e algebra is shown to provide a new Ruijsenaars-Schneider-like Hamiltonian.

A simultaneous and global scheme of quantum deformation is defined for the set of algebras corresponding to the groups of motions of the two-dimensional Cayley-Klein geometries. Their central extensions are also considered under this unified pattern. In both cases some fundamental properties characterizing the classical CK geometries (as the existence of a set of commuting involutions, contractions and dualities relationships), remain in the quantum version.