  The presented thesis is devoted to the study of instabilities of compact
objects within the Einstein-Gauss-Bonnet theory. This theory includes
higher-order corrections in curvature, which are inspired by the low energy
limit of string theory. We study linear instability of higher-dimensional black
holes in the de Sitter universe. The time-domain picture allows us to obtain
the parametric region of stability for the gravitational perturbations in all
three channels, i.e., for scalar-type, vector-type, and tensor-type
perturbations. We observed that while the scalar and tensor channels show
instability for some choice of the parameters, the vector-type perturbations
are always stable. Furthermore, we show that the quasinormal frequencies of the
scalar type of gravitational perturbations do not obey Hod's inequality,
however, the other two channels, vector and tensor, have lower-lying modes that
confirm Hod's conjecture. We also studied stability of the wormholes in the
four-dimensional Einstein-dilaton-Gauss-Bonnet gravity proposed by P. Kanti, B.
Kleihaus, J. Kunz in arXiv:1108.3003 . These wormholes were claimed to be
stable against linear radial perturbations. However, our time-domain analysis
allowed us to prove such wormholes are linearly unstable against general radial
perturbations for any values of their parameters. We observed that the
exponential growth appears after a long period of damped oscillations. This
behaviour is qualitatively similar to the instability profile of the
higher-dimensional black holes in the Einstein-Gauss-Bonnet theory.
