  We study the stability of static black holes in generalized
Einstein-Maxwell-scalar theories. We derive the master equations for the odd
and even parity perturbations. The sufficient and necessary conditions for the
stability of black holes under odd-parity perturbations are derived. We show
that these conditions are usually not similar to energy conditions even in the
simplest case of a minimally coupled scalar field. We obtain the necessary
conditions for the stability of even-parity perturbations. We also derived the
speed of propagation of the five degrees of freedom and obtained the class of
theories for which all degrees of freedom propagate at the speed of light.
Finally, we have applied our results to various black holes in nonlinear
electrodynamics, scalar-tensor theories and Einstein-Maxwell-dilaton theory.
For the latter, we have also calculated the quasinormal modes.

General Relativity and Quantum Cosmology;High Energy Physics - Theory