The superintegrability of several Hamiltonian systems defined on three-dimensional configuration spaces of constant curvature is studied. We first analyze the properties of the Killing vector fields, Noether symmetries and Noether momenta. Then we study the superintegrability of the Harmonic Oscillator, the Smorodinsky-Winternitz (S-W) system and the Harmonic Oscillator with ratio of frequencies 1:1:2 and additional nonlinear terms on the 3-dimensional sphere $S^3$ ($kp>0)$ and on the hyperbolic space $H^3$ ($kp<0$). In the second part we present a study first of the Kepler problem and then of the Kepler problem with additional nonlinear terms in these two curved spaces, $S^3$ ($kp>0)$ and $H^3$ ($kp<0$). We prove their superintegrability and we obtain, in all the cases, the maximal number of functionally independent integrals of motion. All the mathematical expressions are presented using the curvature $kp$ as a parameter, in such a way that particularizing for $kp>0$, $kp=0$, or $kp<0$, the corresponding properties are obtained for the system on the sphere $S^3$, the Euclidean space $IE^3$, or the hyperbolic space $H^3$, respectively.
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