The construction of completely integrable systems from symplectic realizations of Pois son coalgebras with Casimir elements is reviewed [1].Several examples of Hamiltonians with either undeformed or quantum coalgebra symmetry are given, and this symmetry is shown to provide a universal prescription for the choice of the dynamical variables [2,3].The quasi-maximal Liouville superintegrability of any coalgebra model is discussed and the essential role of symplectic realizations in this context is emphasized [4].As a concrete application, sl(2)-coalgebra kinetic energy Hamiltonians describing geodesic motions are shown to generate "dynamically" a large family of N-dimensional curved spaces.More over, certain potentials on these ND spaces can be also introduced by making use of the coalgebra symmetry, in such a way that the integrability properties of the full system are preserved [5,6].In particular the first example of a maximally superintegrable system on a ND space of non-constant curvature is presented [7].