  We discuss the leading order correction to the equation of motion of a
particle with spin on an arbitrary spacetime. A particle traveling in a curved
spacetime is known to trace a geodesic of the background spacetime if the mass
of the particle is negligibly small. For a spinning particle, it is known that
there appears a term due to the coupling of the spin and the Riemann tensor of
the background spacetime. Recently we have found the equation of motion of a
non-spinning particle which includes the effect of gravitational radiation
reaction. This paper is devoted to discussion of a consistent derivation of the
equation of motion which is corrected both by the spin-Riemann coupling and the
gravitational radiation reaction.
