  We review our recent work on linear stability for scalar perturbations of
Kerr spacetimes, that is to say, boundedness and decay properties for solutions
of the scalar wave equation \Box_g{\psi} = 0 on Kerr exterior backgrounds. We
begin with the very slowly rotating case |a| \ll M, where first boundedness and
then decay has been shown in rapid developments over the last two years,
following earlier progress in the Schwarzschild case a = 0. We then turn to the
general subextremal range |a| < M, where we give here for the first time the
essential elements of a proof of definitive decay bounds for solutions {\psi}.
These developments give hope that the problem of the non-linear stability of
the Kerr family of black holes might soon be addressed. This paper accompanies
a talk by one of the authors (I.R.) at the 12th Marcel Grossmann Meeting,
Paris, June 2009.
