In this talk I will present a general framework to establish the stability of  inequalities of the form <Lu, u> >= F(u); where L is a positive linear operator and F is a 2-homogeneous nonlinear functional.   We will then see how this framework can be employed to obtain some stability results concerning the fractional Caffarelli-Kohn-Niremberg inequality. This is joint work with N. De Nitti and T. Konig.