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The superintegrability of several Hamiltonian systems defined onthree-dimensional configuration spaces of constant curvature is studied. Wefirst analyze the properties of the Killing vector fields, Noether symmetriesand Noether momenta. Then we study the superintegrability of the HarmonicOscillator, the Smorodinsky-Winternitz (S-W) system and the Harmonic Oscillatorwith ratio of frequencies 1:1:2 and additional nonlinear terms on the3-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 problemand then of the Kepler problem with additional nonlinear terms in these twocurved spaces, $S^3$ ($\kp>0)$ and $H^3$ ($\kp<0$). We prove theirsuperintegrability and we obtain, in all the cases, the maximal number offunctionally independent integrals of motion.  All the mathematical expressions are presented using the curvature $\kp$ as aparameter, 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.

Superintegrability on the 3-dimensional spaces with curvature.  Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the  Hyperbolic space $H^3$