In this talk I shall discuss my recent work which establishes that the Schwarzschild family of black holes are linearly stable as a family of solutions to the system of equations that result from expressing the Einstein vacuum equations in a generalised wave gauge. This result therefore provides an important step towards a resolution of the black hole stability problem in general relativity and thus complements the recent work of Dafermos—Holzegel—Rodnianski in a similar vein as to how the work of Lindblad—Rodnianski complemented that of Christodoulou—Klainerman in establishing the non-linear stability of the Minkowski space.