  Black holes have their proper oscillations, which are called the quasi-normal
modes. The proper oscillations of astrophysical black holes can be observed in
the nearest future with the help of gravitational wave detectors. Quasi-normal
modes are also very important in the context of testing of the stability of
black objects, the anti-de Sitter/Conformal Field Theory (AdS/CFT)
correspondence and in higher dimensional theories, such as the brane-world
scenarios and string theory.
  This dissertation reviews a number of works, which provide a thorough study
of the quasi-normal spectrum of a wide class of black holes in four and higher
dimensions for fields of various spin and gravitational perturbations. We have
studied numerically the dependance of the quasi-normal modes on a number of
factors, such as the presence of the cosmological constant, the Gauss-Bonnet
parameter or the aether in the space-time, the dependance of the spectrum on
parameters of the black hole and fields under consideration. By the analysis of
the quasi-normal spectrum, we have studied the stability of higher dimensional
Reissner-Nordstrom-de Sitter black holes, Kaluza-Klein black holes with
squashed horizons, Gauss-Bonnet black holes and black strings. Special
attention is paid to the evolution of massive fields in the background of
various black holes. We have considered their quasi-normal ringing and the
late-time tails. In addition, we present two new numerical techniques: a
generalisation of the Nollert improvement of the Frobenius method for higher
dimensional problems and a qualitatively new method, which allows to calculate
quasi-normal frequencies for black holes, which metrics are not known
analytically.
