  This paper concludes the series begun in [M. Dafermos and I. Rodnianski,
Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the
cases |a| << M or axisymmetry, arXiv:1010.5132], providing the complete proof
of definitive boundedness and decay results for the scalar wave equation on
Kerr backgrounds in the general subextremal |a| < M case without symmetry
assumptions. The essential ideas of the proof (together with explicit
constructions of the most difficult multiplier currents) have been announced in
our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for
linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann
Meeting on General Relativity, T. Damour et al (ed.), World Scientific,
Singapore, 2011, pp. 132-189, arXiv:1010.5137]. Our proof appeals also to the
quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative
Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to
appear, Ann. Henri Poincare], together with a streamlined continuity argument
in the parameter a, appearing here for the first time. While serving as Part
III of a series, this paper repeats all necessary notations so that it can be
read independently of previous work.
