  The harmonic oscillator as a distinguished dynamical system can be defined
not only on the Euclidean plane but also on the sphere and on the hyperbolic
plane, and more generally on any configuration space with constant curvature
and with a metric of any signature, either Riemannian (definite positive) or
Lorentzian (indefinite). In this paper we study the main properties of these
`curved' harmonic oscillators simultaneously on any such configuration space,
using a Cayley-Klein (CK) type approach, with two free parameters $\ki, \kii$
which altogether correspond to the possible values for curvature and signature
type: the generic Riemannian and Lorentzian spaces of constant curvature
(sphere ${\bf S}^2$, hyperbolic plane ${\bf H}^2$, AntiDeSitter sphere ${\bf
AdS}^{\unomasuno}$ and DeSitter sphere ${\bf dS}^{\unomasuno}$) appear in this
family, with the Euclidean and Minkowski spaces as flat limits.
  We solve the equations of motion for the `curved' harmonic oscillator and
obtain explicit expressions for the orbits by using three different methods:
first by direct integration, second by obtaining the general CK version of the
Binet's equation and third, as a consequence of its superintegrable character.
The orbits are conics with centre at the potential origin in any CK space,
thereby extending this well known Euclidean property to any constant curvature
configuration space. The final part of the article, that has a more geometric
character, presents those results of the theory of conics on spaces of constant
curvature which are pertinent.
