  We derive the large time asymptotics of initially regular and localized
solutions of the Teukolsky equation on the exterior of a subextremal Kerr black
hole for any half integer spin. More precisely, we obtain the leading order
term (predicted by Price's law) in the large time regime assuming that the
initial data have compact support and have enough (but finite) Sobolev
regularity. For initial data with less spatial decay (typically decaying like
r^{--1--$\alpha$} with $\alpha$ $\in$ (0, 1)), we prove that the solution has a
pointwise decay of order t^{--1--$\alpha$--s--|s|+} on spatially compact
regions. In the proof, we adopt the spectral point of view and make use of
recent advances in microlocal analysis and non elliptic Fredholm theory which
provide a robust framework to study linear operators on black hole type
spacetimes.
