  The Kepler problem is a dynamical system that is well defined not only on the
Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the
theory of central potentials on spaces of constant curvature is studied. All
the mathematical expressions are presented using the curvature $\k$ as a
parameter, in such a way that they reduce to the appropriate property for the
system on the sphere $S^2$, or on the hyperbolic plane $H^2$, when
particularized for $\k>0$, or $\k<0$, respectively; in addition, the Euclidean
case arises as the particular case $\k=0$. In the second part we study the main
properties of the Kepler problem on spaces with curvature, we solve the
equations and we obtain the explicit expressions of the orbits by using two
different methods: first by direct integration and second by obtaining the
$\k$-dependent version of the Binet's equation. The final part of the article,
that has a more geometric character, is devoted to the study of the theory of
conics on spaces of constant curvature.
