  The concept of elementary particle rests on the idea that it is a physical
system with no excited states, so that all possible kinematical states of the
particle are just kinematical modifications of any one of them. The way of
describing the particle attributes is equivalent to describe the collection of
consecutive inertial observers who describe the particle in the same
kinematical state. The kinematical state space of an elementary particle is a
homogeneous space of the kinematical group. By considering the largest
homogeneous spaces of both, Galilei and Poincar\'e groups, it is shown how the
spin structure is related to the different degrees of freedom. The formalism is
quantized by means of Feynman's path integral approach and special attention is
paid to the classical model which satisfies Dirac's equation. Dirac's algebra
is related to the classical observables, in particular to the orientation
variables. Several spin effects are also analyzed.
